Sequences, Series, and Probability 9 - sciannamath.weebly.comsciannamath.weebly.com/uploads/1/6/...sequences_series_and_probability.pdf · •Use sequences and series to model and

Poker has become a popular cardgame in recent years. You can use the probability theory developed in this chapter to calculate the likelihood of getting different pokerhands.

641

SELECTED APPLICATIONSSequences, series, and probability have many real-life applications. The applications listed below rep-resent a small sample of the applications in this chapter.

• Federal Debt,Exercise 111, page 651

• Falling Object,Exercises 87 and 88, page 661

• Multiplier Effect,Exercises 113–116, page 671

• Data Analysis: Tax Returns,Exercise 61, page 682

• Child Support,Exercise 80, page 690

• Poker Hand Exercise 57, page 699

• Lottery,Exercise 65, page 700

• Defective Units,Exercise 47, page 711

• Population Growth,Exercise 139, page 718

9.1 Sequences and Series

9.2 Arithmetic Sequences and Partial Sums

9.3 Geometric Sequences and Series

9.4 Mathematical Induction

9.5 The Binomial Theorem

9.6 Counting Principles

9.7 Probability

Sequences, Series, and Probability 99

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333202_0900.qxd 12/5/05 11:26 AM Page 641

SequencesIn mathematics, the word sequence is used in much the same way as in ordinaryEnglish. Saying that a collection is listed in sequence means that it is ordered sothat it has a first member, a second member, a third member, and so on.

Mathematically, you can think of a sequence as a function whose domain isthe set of positive integers.

Rather than using function notation, however, sequences are usually writtenusing subscript notation, as indicated in the following definition.

On occasion it is convenient to begin subscripting a sequence with 0 insteadof 1 so that the terms of the sequence become

Writing the Terms of a Sequence

Write the first four terms of the sequences given by

a. b.

Solution

a. The first four terms of the sequence given by are

1st term

2nd term

3rd term

4th term

b. The first four terms of the sequence given by are

1st term

2nd term

3rd term

4th term

Now try Exercise 1.

a4 � 3 � ��1�4 � 3 � 1 � 4.

a3 � 3 � ��1�3 � 3 � 1 � 2

a2 � 3 � ��1�2 � 3 � 1 � 4

a1 � 3 � ��1�1 � 3 � 1 � 2

an � 3 � ��1�n

a4 � 3�4� � 2 � 10.

a3 � 3�3� � 2 � 7

a2 � 3�2� � 2 � 4

a1 � 3�1� � 2 � 1

an � 3n � 2

an � 3 � ��1�n.an � 3n � 2

a0, a1, a2, a3, . . . .

. . .f �4� � a4, . . . , f �n� � an,f �3� � a3,f �2� � a2,f�1� � a1,

642 Chapter 9 Sequences, Series, and Probability

What you should learn• Use sequence notation to

write the terms of sequences.

• Use factorial notation.

• Use summation notation towrite sums.

• Find the sums of infinite series.

• Use sequences and series tomodel and solve real-lifeproblems.

Why you should learn itSequences and series can beused to model real-life problems.For instance, in Exercise 109 onpage 651, sequences are used tomodel the number of Best Buystores from 1998 through 2003.

Sequences and Series

Scott Olson /Getty Images

9.1

Definition of SequenceAn infinite sequence is a function whose domain is the set of positiveintegers. The function values

. . . , . . .

are the terms of the sequence. If the domain of the function consists of thefirst positive integers only, the sequence is a finite sequence.n

an,a4,a3,a2,a1,

Example 1

The HM mathSpace® CD-ROM andEduspace® for this text contain additional resources related to the concepts discussed in this chapter.

333202_0901.qxd 12/5/05 11:28 AM Page 642

A Sequence Whose Terms Alternate in Sign

Write the first five terms of the sequence given by

SolutionThe first five terms of the sequence are as follows.

1st term

2nd term

3rd term

4th term

5th term

Now try Exercise 17.

Simply listing the first few terms is not sufficient to define a uniquesequence—the th term must be given. To see this, consider the followingsequences, both of which have the same first three terms.

Finding the nth Term of a Sequence

Write an expression for the apparent th term of each sequence.

a. b.

Solutiona. n: 1 2 3 4

Terms: 1 3 5 7 Apparent pattern: Each term is 1 less than twice which implies that

b. n: 1 2 3 4Terms: 2 10 Apparent pattern: The terms have alternating signs with those in the evenpositions being negative. Each term is 1 more than the square of whichimplies that


an � ��1�n�1�n2 � 1�

n,

. . . an�17�5

. . . n

an � 2n � 1.

n,. . . an

. . . n

2, �5, 10, �17, . . .1, 3, 5, 7, . . .

�an �n

1

2,

1

4,

1

8,

1

15, . . . ,

6

�n � 1��n2 � n � 6�, . . .

1

2,

1

4,

1

8,

1

16, . . . ,

1

2n, . . .

n

a5 ��1�5

2�5� � 1�

�1

10 � 1� �

1

9

a4 ��1�4

2�4� � 1�

1

8 � 1�

1

7

a3 ��1�3

2�3� � 1�

�1

6 � 1� �

1

5

a2 ��1�2

2�2� � 1�

1

4 � 1�

1

3

a1 ��1�1

2�1� � 1�

�1

2 � 1� �1

an ��1�n

2n � 1.

Section 9.1 Sequences and Series 643

Example 2

Example 3

Write out the first five terms ofthe sequence whose th term is

Are they the same as the firstfive terms of the sequence inExample 2? If not, how do theydiffer?

an ��1�n�1

2n � 1.

n

Exploration

To graph a sequence using agraphing utility, set the mode tosequence and dot and enter thesequence. The graph of thesequence in Example 3(a) isshown below. You can use thetrace feature or value feature toidentify the terms.

00 5

11

Techno logy

333202_0901.qxd 12/5/05 11:28 AM Page 643

Some sequences are defined recursively. To define a sequence recursively,you need to be given one or more of the first few terms. All other terms of thesequence are then defined using previous terms. A well-known example is theFibonacci sequence shown in Example 4.

The Fibonacci Sequence: A Recursive Sequence

The Fibonacci sequence is defined recursively, as follows.

where

Write the first six terms of this sequence.

Solution0th term is given.

1st term is given.

Use recursion formula.





Factorial NotationSome very important sequences in mathematics involve terms that are definedwith special types of products called factorials.

Here are some values of for the first several nonnegative integers. Noticethat is 1 by definition.

The value of does not have to be very large before the value of becomesextremely large. For instance, 10! � 3,628,800.

n!n

5! � 1 � 2 � 3 � 4 � 5 � 120

4! � 1 � 2 � 3 � 4 � 24

3! � 1 � 2 � 3 � 6

2! � 1 � 2 � 2

1! � 1

0! � 1

0!n!

a5 � a5�2 � a5�1 � a3 � a4 � 3 � 5 � 8

a4 � a4�2 � a4�1 � a2 � a3 � 2 � 3 � 5

a3 � a3�2 � a3�1 � a1 � a2 � 1 � 2 � 3

a2 � a2�2 � a2�1 � a0 � a1 � 1 � 1 � 2

a1 � 1

a0 � 1

k ≥ 2ak � ak�2 � ak�1,a1 � 1,a0 � 1,


The subscripts of a sequencemake up the domain of thesequence and they serve to identify the location of a termwithin the sequence. For exam-ple, is the fourth term of thesequence, and is the nth termof the sequence. Any variablecan be used as a subscript. Themost commonly used variablesubscripts in sequence andseries notation are and n.k,j,i,

an

a4

Example 4

Definition of FactorialIf is a positive integer, n factorial is defined as

As a special case, zero factorial is defined as 0! � 1.

n! � 1 � 2 � 3 � 4 . . . �n � 1� � n.

n

333202_0901.qxd 12/5/05 11:28 AM Page 644

Factorials follow the same conventions for order of operations as do expo-nents. For instance,

whereas

Writing the Terms of a Sequence Involving Factorials

Write the first five terms of the sequence given by

Begin with Then graph the terms on a set of coordinate axes.

Solution

0th term

1st term

2nd term

3rd term

4th term

Figure 9.1 shows the first five terms of the sequence.


When working with fractions involving factorials, you will often find thatthe fractions can be reduced to simplify the computations.

Evaluating Factorial Expressions

Evaluate each factorial expression.

a. b. c.

Solution

a.

b.

c.


n!

�n � 1�!�

1 � 2 � 3 . . . �n � 1� � n

1 � 2 � 3 . . . �n � 1�� n

2! � 6!

3! � 5!�

1 � 2 � 1 � 2 � 3 � 4 � 5 � 6

1 � 2 � 3 � 1 � 2 � 3 � 4 � 5�

6

3� 2

8!

2! � 6!�

1 � 2 � 3 � 4 � 5 � 6 � 7 � 8

1 � 2 � 1 � 2 � 3 � 4 � 5 � 6�

7 � 8

2� 28

n!

�n � 1�!2! � 6!

3! � 5!

8!

2! � 6!

a4 �24

4!�

16

24�

2

3

a3 �23

3!�

8

6�

4

3

a2 �22

2!�

4

2� 2

a1 �21

1!�

2

1� 2

a0 �20

0!�

1

1� 1

n � 0.

an �2n

n!.

�2n�! � 1 � 2 � 3 � 4 . . . 2n.

� 2�1 � 2 � 3 � 4 . . . n�2n! � 2�n!�


Example 5

Example 6

n21 3 4

2

4

1

3

an

FIGURE 9.1

Note in Example 6(a) that youcan simplify the computation asfollows.

�8 � 72 � 1

� 28

8!

2! � 6!�

8 � 7 � 6!2! � 6!

333202_0901.qxd 12/5/05 11:28 AM Page 645

Summation NotationThere is a convenient notation for the sum of the terms of a finite sequence. It iscalled summation notation or sigma notation because it involves the use of theuppercase Greek letter sigma, written as

Summation Notation for Sums

Find each sum.

a. b. c.

Solution

a.

b.

c.

For this summation, note that the sum is very close to the irrational numberIt can be shown that as more terms of the sequence whose

th term is are added, the sum becomes closer and closer to


In Example 7, note that the lower limit of a summation does not have to be1. Also note that the index of summation does not have to be the letter Forinstance, in part (b), the letter is the index of summation.k

i.

e.1�n!ne � 2.718281828.

� 2.71828

� 1 � 1 �1

2�

1

6�

1

24�

1

120�

1

720�

1

5040�

1

40,320

�8

i�0

1

i!�

1

0!�

1

1!�

1

2!�

1

3!�

1

4!�

1

5!�

1

6!�

1

7!�

1

8!

� 90

� 10 � 17 � 26 � 37

�6

k�3

�1 � k2� � �1 � 32� � �1 � 42� � �1 � 52� � �1 � 62�

� 45

� 3�15� � 3�1 � 2 � 3 � 4 � 5�

�5

i�1

3i � 3�1� � 3�2� � 3�3� � 3�4� � 3�5�

�8

i�0 1i!�

6

k�3�1 � k2��

5

i�13i

�.


Most graphing utilities are able to sum the first n terms of asequence. Check your user’s guidefor a sum sequence feature or aseries feature.

Techno logy

Summation notation is aninstruction to add the terms ofa sequence. From the definitionat the right, the upper limit ofsummation tells you where toend the sum. Summationnotation helps you generatethe appropriate terms of thesequence prior to finding theactual sum, which may beunclear.

Definition of Summation NotationThe sum of the first terms of a sequence is represented by

where is called the index of summation, is the upper limit ofsummation, and 1 is the lower limit of summation.

ni

�n

i�1

ai � a1 � a2 � a3 � a4 � . . . � an

n

Example 7

333202_0901.qxd 12/5/05 11:28 AM Page 646


Variations in the upper andlower limits of summation canproduce quite different-lookingsummation notations for thesame sum. For example, thefollowing two sums have thesame terms.

�2

i�03�2i�1� � 3�21 � 22 � 23�

�3

i�13�2i� � 3�21 � 22 � 23�

For proofs of these properties, see Proofs in Mathematics on page 722.

SeriesMany applications involve the sum of the terms of a finite or infinite sequence.Such a sum is called a series.

Finding the Sum of a Series

For the series find (a) the third partial sum and (b) the sum.

Solutiona. The third partial sum is

b. The sum of the series is


� 0.33333. . . �13

.

� 0.3 � 0.03 � 0.003 � 0.0003 � 0.00003 � . . .

��

i�1

310i �

3101 �

3102 �

3103 �

3104 �

3105 � . . .

�3

i�1

310i �

3101 �

3102 �

3103 � 0.3 � 0.03 � 0.003 � 0.333.

��

i�1

310i ,

Properties of Sums

1. is a constant. 2. is a constant.

3. 4. �n

i�1

�ai � bi� � �n

i�1

ai � �n

i�1

bi�n

i�1

�ai � bi� � �n

i�1

ai � �n

i�1

bi

c�n

i�1

cai � c�n

i�1

ai,c�n

i�1

c � cn,

Definition of SeriesConsider the infinite sequence

1. The sum of the first terms of the sequence is called a finite series orthe nth partial sum of the sequence and is denoted by

2. The sum of all the terms of the infinite sequence is called an infiniteseries and is denoted by

a1 � a2 � a3 � . . . � ai � . . . � ��

i�1ai.

a1 � a2 � a3 � . . . � an � �n

i�1ai.

n

. . . .ai ,. . . ,a3,a2,a1,

Example 8

333202_0901.qxd 12/5/05 11:28 AM Page 647

ApplicationSequences have many applications in business and science. One such applicationis illustrated in Example 9.

Population of the United States

For the years 1980 to 2003, the resident population of the United States can beapproximated by the model

where is the population (in millions) and represents the year, with corresponding to 1980. Find the last five terms of this finite sequence, whichrepresent the U.S. population for the years 1999 to 2003. (Source: U.S. CensusBureau)

SolutionThe last five terms of this finite sequence are as follows.

1999 population

2000 population

2001 population

2002 population

2003 population


a23 � 226.9 � 2.05�23� � 0.035�23�2 � 292.6

a22 � 226.9 � 2.05�22� � 0.035�22�2 � 288.9

a21 � 226.9 � 2.05�21� � 0.035�21�2 � 285.4

a20 � 226.9 � 2.05�20� � 0.035�20�2 � 281.9

a19 � 226.9 � 2.05�19� � 0.035�19�2 � 278.5

n � 0nan

n � 0, 1, . . . , 23an � 226.9 � 2.05n � 0.035n2,


Example 9

A cube is created using 27 unit cubes (a unit cube has a length,width, and height of 1 unit) and only the faces of each cube that are visibleare painted blue (see Figure 9.2). Complete the table below to determine howmany unit cubes of the cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. Do the same for a cube, a cube,and a cube and add your results to the table below. What type ofpattern do you observe in the table? Write a formula you could use to determinethe column values for an cube.n � n � n

6 � 6 � 65 � 5 � 54 � 4 � 4

3 � 3 � 3

3 � 3 � 3

Exploration

FIGURE 9.2Number of

0 1 2 3blue cube faces

3 � 3 � 3

333202_0901.qxd 12/5/05 11:28 AM Page 648


In Exercises 1–22, write the first five terms of the sequence.(Assume that begins with 1.)

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

In Exercises 23–26, find the indicated term of the sequence.

23. 24.

25. 26.

In Exercises 27–32, use a graphing utility to graph the first10 terms of the sequence. (Assume that begins with 1.)

27. 28.

29. 30.

31. 32.

In Exercises 33–36, match the sequence with the graph ofits first 10 terms. [The graphs are labeled (a), (b), (c), and(d).]

(a) (b)

(c) (d)

33. 34.

35. 36. an �4n

n!an � 4�0.5�n�1

an �8n

n � 1an �

8

n � 1

2 106 84

2

4

6

8

10

n

an

2 106 84

2

4

6

8

10

n

an

2 106 84

2

4

6

8

10

n

an

2 106 84

2

4

6

8

10

n

an

an �n2

n2 � 2an �

2n

n � 1

an � 8�0.75�n�1an � 16��0.5�n�1

an � 2 �4

nan �

3

4n

n

a13 � �a11 � �

an �4n2 � n � 3

n�n � 1��n � 2�an �

4n

2n2 � 3

a16 � �a25 � �an � ��1�n�1�n�n � 1��an � ��1�n�3n � 2�

an � n�n2 � 6�an � n�n � 1��n � 2�an � 0.3an �

23

an � ��1�n� n

n � 1an ��1�n

n2

an �10

n2�3an �

1

n3�2

an �2n

3nan � 2 �

1

3n

an � 1 � ��1�nan �1 � ��1�n

n

an �3n2 � n � 4

2n2 � 1an �

6n

3n 2 � 1

an �n

n � 2an �

n � 2

n

an � ��12�n

an � ��2�n

an � �12�n

an � 2n

an � 5n � 3an � 3n � 1

n

Exercises 9.1 The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help.

VOCABULARY CHECK: Fill in the blanks.

1. An ________ ________ is a function whose domain is the set of positive integers.2. The function values are called the ________ of a sequence.3. A sequence is a ________ sequence if the domain of the function consists of the first positive integers.4. If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are

defined using previous terms, then the sequence is said to be defined ________.

5. If is a positive integer, ________ is defined as

6. The notation used to represent the sum of the terms of a finite sequence is ________ ________ or sigma notation.

7. For the sum is called the ________ of summation, is the ________ limit of summation, and 1 is

the ________ limit of summation.

8. The sum of the terms of a finite or infinite sequence is called a ________.

9. The ________ ________ ________ of a sequence is the sum of the first terms of the sequence.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

n

ni�n

i�1 ai ,

n! � 1 � 2 � 3 � 4 . . . �n � 1� � n.nn

na1, a2, a3, a4, . . .

333202_0901.qxd 12/5/05 11:28 AM Page 649

www.Eduspace.com

In Exercises 37–50, write an expression for the apparent th term of the sequence. (Assume that begins with 1.)

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49.

50.

In Exercises 51–54, write the first five terms of the sequencedefined recursively.

51.

52.

53.

54.

In Exercises 55–58, write the first five terms of the sequencedefined recursively. Use the pattern to write the th term of the sequence as a function of . (Assume that beginswith 1.)

55.

56.

57.

58.

In Exercises 59–64, write the first five terms of thesequence. (Assume that begins with 0.)

59. 60.

61. 62.

63. 64.

In Exercises 65–72, simplify the factorial expression.

65. 66.

67. 68.

69. 70.

71. 72.

In Exercises 73–84, find the sum.

73. 74.

75. 76.

77. 78.

79. 80.

81. 82.

83. 84.

In Exercises 85–88, use a calculator to find the sum.

85. 86.

87.

88.

In Exercises 89–98, use sigma notation to write the sum.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

In Exercises 99–102, find the indicated partial sum of theseries.

99. 100.

Fourth partial sum Fifth partial sum

101. 102.

Third partial sum Fourth partial sum

��

n�18��1

4�n��

n�14��1

2�n

��

i�12�1

3�i

��

i�15�1

2�i

12 �

24 �

68 �

2416 �

12032 �

72064

14 �

38 �

716 �

1532 �

3164

1

1 � 3�

1

2 � 4�

1

3 � 5� . . . �

1

10 � 12

1

12�

1

22�

1

32�

1

42� . . . �

1

202

1 �12 �

14 �

18 � . . . �

1128

3 � 9 � 27 � 81 � 243 � 729

�1 � �16�2� � �1 � �2

6�2� � . . . � �1 � �66�2�

�2� 18� � 3� � �2�28� � 3� � . . . � �2� 88� � 3�

5

1 � 1�

5

1 � 2�

5

1 � 3� . . . �

5

1 � 15

1

3�1��

1

3�2��

1

3�3�� . . . �

1

3�9�

�4

k�0

��1�k

k!

�4

k�0

��1�k

k � 1

�10

j�1

3

j � 1�6

j�1

�24 � 3j�

�4

j�0

��2� j�4

i�1

2i

�4

i�1

��i � 1�2 � �i � 1�3��5

k�2

�k � 1�2�k � 3�

�5

j�3

1

j 2 � 3�3

k�0

1

k2 � 1

�5

i�0

2i 2�4

i�0

i 2

�5

k�1

5�4

k�1

10

�6

i�1

�3i � 1��5

i�1

�2i � 1�

�3n � 1�!�3n�!

�2n � 1�!�2n � 1�!

�n � 2�!n!

�n � 1�!n!

25!

23!

10!

8!

5!

8!

4!

6!

an ��1�2n�1

�2n � 1�!an ��1�2n

�2n�!

an �n2

�n � 1�!an �1

�n � 1�!

an �n!n

an �3n

n!

n

ak�1 � ��2�aka1 � 14,

ak�1 �13aka1 � 81,

ak�1 � ak � 5a1 � 25,

ak�1 � ak � 2a1 � 6,

nnn

ak�1 �12aka1 � 32,

ak�1 � 2�ak � 1�a1 � 3,

ak�1 � ak � 3a1 � 15,

ak�1 � ak � 4a1 � 28,

1 �12, 1 �

34, 1 �

78, 1 �

1516, 1 �

3132, . . .

1 �11, 1 �

12, 1 �

13, 1 �

14, 1 �

15, . . .

1, 2, 22

2,

23

6,

24

24,

25

120, . . .1, �1, 1, �1, 1, . . .

1, 12, 16, 124, 1

120, . . .1, 14, 19, 116,

125, . . .

13, 29, 4

27, 881, . . .2

1, 33, 45, 57, 69, . . .

12, �1

4 , 18, �116 , . . .�2

3 , 34, �45 , 56, �6

7 , . . .

2, �4, 6, �8, 10, . . .0, 3, 8, 15, 24, . . .

3, 7, 11, 15, 19, . . .1, 4, 7, 10, 13, . . .

nn


333202_0901.qxd 12/5/05 11:28 AM Page 650


In Exercises 103–106, find the sum of the infinite series.

103.

104.

105.

106.

107. Compound Interest A deposit of $5000 is made in anaccount that earns 8% interest compounded quarterly.The balance in the account after quarters is given by

(a) Write the first eight terms of this sequence.

(b) Find the balance in this account after 10 years byfinding the 40th term of the sequence.

108. Compound Interest A deposit of $100 is made eachmonth in an account that earns 12% interest compoundedmonthly. The balance in the account after months isgiven by

(a) Write the first six terms of this sequence.

(b) Find the balance in this account after 5 years byfinding the 60th term of the sequence.

(c) Find the balance in this account after 20 years byfinding the 240th term of the sequence.

110. Medicine The numbers (in thousands) of AIDS casesreported from 1995 to 2003 can be approximated by themodel

where is the year, with corresponding to 1995.(Source: U.S. Centers for Disease Control andPrevention)

(a) Find the terms of this finite sequence. Use the statisticalplotting feature of a graphing utility to construct a bargraph that represents the sequence.

(b) What does the graph in part (a) say about reportedcases of AIDS?

111. Federal Debt From 1990 to 2003, the federal debt of theUnited States rose from just over $3 trillion to almost $7trillion. The federal debt (in billions of dollars) from1990 to 2003 is approximated by the model

where is the year, with corresponding to 1990.(Source: U.S. Office of Management and Budget)

(a) Find the terms of this finite sequence. Use the statisticalplotting feature of a graphing utility to construct a bargraph that represents the sequence.

(b) What does the pattern in the bar graph in part (a) sayabout the future of the federal debt?

n � 0n

n � 0, 1, . . . , 13an � 2.7698n3 � 61.372n2 � 600.00n � 3102.9,

an

n � 5n

n � 5, 6, . . . , 13an � 0.0457n3 � 0.352n2 � 9.05n � 121.4,

an

n � 1, 2, 3, . . . .An � 100�101��1.01�n � 1� ,

n

n � 1, 2, 3, . . . .An � 5000�1 �0.08

4 n

,

n

��

i�12� 1

10�i

��

k�17� 1

10�k

��

k�1� 110�k

��

i�16� 1

10�i

109. Data Analysis: Number of Stores The table showsthe numbers of Best Buy stores for the years 1998to 2003. (Source: Best Buy Company, Inc.)

an

Model It

Year Number ofstores,

1998 311

1999 357

2000 419

2001 481

2002 548

2003 608

an

Model It (cont inued)

(a) Use the regression feature of a graphing utility tofind a linear sequence that models the data. Let represent the year, with corresponding to1998.

(b) Use the regression feature of a graphing utility tofind a quadratic sequence that models the data.

(c) Evaluate the sequences from parts (a) and (b) forCompare these values with

those shown in the table. Which model is a better fitfor the data? Explain.

(d) Which model do you think would better predict thenumber of Best Buy stores in the future? Use themodel you chose to predict the number of Best Buystores in 2008.

n � 8, 9, . . . , 13.

n � 8n

333202_0901.qxd 12/5/05 11:28 AM Page 651

112. Revenue The revenues (in millions of dollars) forAmazon.com for the years 1996 through 2003 are shownin the figure. The revenues can be approximated by themodel

where is the year, with corresponding to 1996.Use this model to approximate the total revenue from1996 through 2003. Compare this sum with the result ofadding the revenues shown in the figure. (Source:Amazon.com)

Synthesis

True or False? In Exercises 113 and 114, determinewhether the statement is true or false. Justify your answer.

113. 114.

Fibonacci Sequence In Exercises 115 and 116, use theFibonacci sequence. (See Example 4.)

115. Write the first 12 terms of the Fibonacci sequence andthe first 10 terms of the sequence given by

116. Using the definition for in Exercise 115, show that can be defined recursively by

Arithmetic Mean In Exercises 117–120, use the followingdefinition of the arithmetic mean of a set of measurements

117. Find the arithmetic mean of the six checking accountbalances $327.15, $785.69, $433.04, $265.38, $604.12,and $590.30. Use the statistical capabilities of a graphingutility to verify your result.

118. Find the arithmetic mean of the following prices pergallon for regular unleaded gasoline at five gasolinestations in a city: $1.899, $1.959, $1.919, $1.939, and$1.999. Use the statistical capabilities of a graphingutility to verify your result.

119. Proof Prove that

120. Proof Prove that

In Exercises 121–124, find the first five terms of thesequence.

121. 122.

123. 124.

Skills Review

In Exercises 125–128, determine whether the function hasan inverse function. If it does, find its inverse function.

125. 126.

127. 128.

In Exercises 129–132, find (a) (b) (c)and (d)

129.

130.

131.

132.

In Exercises 133–136, find the determinant of the matrix.

133. 134.

135.

136. A � 169

�2�4

118

�16

103

122

2731�

A � 304

479

53

�1�A � �2

128

15�A � 3�1

57�

B � 03

�1

410

0�2

2�A � �1

50

41

�1

023�,

B � 100

413

261�A �

�241

�357

674�,B � 0

8�12

11�A � 10�4

76�,

B � �26

4�3�A � 6

354�,

BA.AB,4B � 3A,A � B,

f �x� � �x � 1�2h�x� � �5x � 1

g�x� �3x

f �x� � 4x � 3

an ��1�n x2n�1

�2n � 1�!an ��1�n x2n

�2n�!

an ��1�n x2n�1

2n � 1an �

xn

n!

�n

i�1

�xi � x � 2 � �n

i�1

x 2i �

1

n��n

i�1

xi2

.

�n

i�1

�xi � x � � 0.

x �1n

�n

i�1

xi

. . . , xn.x3,x2,x1,nx

bn � 1 �1

bn�1

.

bnbn

n ≥ 1.bn �an�1

an

,

an

�4

j�12 j � �

6

j�32 j�2�

4

i�1�i2 � 2i� � �

4

i�1i 2 � 2�

4

i�1i

Rev

enue

(in

mill

ions

of

dolla

rs)

Year (6 ↔ 1996)

n

an

6 7 8 9 10 11 12 13

1000

2000

3000

4000

5000

6000

n � 6n

n � 6, 7, . . . , 13an � 46.609n2 � 119.84n � 1125.8,

an


333202_0901.qxd 12/5/05 11:28 AM Page 652

Section 9.2 Arithmetic Sequences and Partial Sums 653

What you should learn• Recognize, write, and find

the nth terms of arithmeticsequences.

• Find nth partial sums of arithmetic sequences.

• Use arithmetic sequences tomodel and solve real-life problems.

Why you should learn itArithmetic sequences havepractical real-life applications.For instance, in Exercise 83 on page 660, an arithmeticsequence is used to model theseating capacity of an auditorium.

Arithmetic Sequences and Partial Sums

© mediacolor’s Alamy

9.2

Arithmetic SequencesA sequence whose consecutive terms have a common difference is called anarithmetic sequence.

Examples of Arithmetic Sequences

a. The sequence whose th term is is arithmetic. For this sequence, thecommon difference between consecutive terms is 4.

Begin with

b. The sequence whose th term is is arithmetic. For this sequence, thecommon difference between consecutive terms is

Begin with

c. The sequence whose th term is is arithmetic. For this sequence, thecommon difference between consecutive terms is

Begin with

Now try Exercise 1.

The sequence 1, 4, 9, 16, whose th term is is not arithmetic. Thedifference between the first two terms is

but the difference between the second and third terms is

a3 � a2 � 9 � 4 � 5.

a2 � a1 � 4 � 1 � 3

n2,n. . . ,

54 � 1 �

14

n � 1.1, 5

4,

3

2,

7

4, . . . ,

n � 3

4, . . .

14.

14�n � 3�n

�3 � 2 � �5

n � 1.2, �3, �8, �13, . . . , 7 � 5n, . . .

�5.7 � 5nn

11 � 7 � 4

n � 1.7, 11, 15, 19, . . . , 4n � 3, . . .

4n � 3n

Definition of Arithmetic SequenceA sequence is arithmetic if the differences between consecutive terms arethe same. So, the sequence

is arithmetic if there is a number such that

The number is the common difference of the arithmetic sequence.d

a2 � a1 � a3 � a2 � a4 � a 3 � . . . � d.

d

a1, a2, a3, a4, . . . , an, . . .

Example 1

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In Example 1, notice that each of the arithmetic sequences has an th termthat is of the form where the common difference of the sequence is Anarithmetic sequence may be thought of as a linear function whose domain is theset of natural numbers.

Finding the nth Term of an Arithmetic Sequence

Find a formula for the th term of the arithmetic sequence whose commondifference is 3 and whose first term is 2.

SolutionBecause the sequence is arithmetic, you know that the formula for the th termis of the form Moreover, because the common difference is the formula must have the form

Substitute 3 for

Because it follows that

Substitute 2 for and 3 for

So, the formula for the th term is

The sequence therefore has the following form.


Another way to find a formula for the th term of the sequence in Example2 is to begin by writing the terms of the sequence.

From these terms, you can reason that the th term is of the form

an � dn � c � 3n � 1.

n

a1

2

2

a2

2 � 3

5

a3

5 � 3

8

a4

8 � 3

11

a5

11 � 3

14

a6

14 � 3

17

a7

17 � 3

20

. . .

. . .

. . .

n

2, 5, 8, 11, 14, . . . , 3n � 1, . . .

an � 3n � 1.

n

� �1.

d.a1 � 2 � 3

c � a1 � d

a1 � 2,

d.an � 3n � c.

d � 3,an � dn � c.n

n

d.dn � c,n


a1

c

an = dn + c

n

an

a2 a3

FIGURE 9.3

The alternative recursion formof the th term of an arithmeticsequence can be derived fromthe pattern below.

1st term

2nd term

3rd term

4th term

5th term

1 less

nth term

1 less

an � a1 � �n � 1� d�

a5 � a1 � 4d

a4 � a1 � 3d

a3 � a1 � 2d

a2 � a1 � d

a1 � a1

n

The nth Term of an Arithmetic SequenceThe th term of an arithmetic sequence has the form

Linear form

where is the common difference between consecutive terms of thesequence and A graphical representation of this definition isshown in Figure 9.3. Substituting for in yields analternative recursion form for the nth term of an arithmetic sequence.

Alternative forman � a1 � �n � 1� d

an � dn � cca1 � dc � a1 � d.

d

an � dn � c

n

Example 2

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Writing the Terms of an Arithmetic Sequence

The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Writethe first 11 terms of this sequence.

SolutionYou know that and So, you must add the common difference

nine times to the fourth term to obtain the 13th term. Therefore, the fourth and13th terms of the sequence are related by

and are nine terms apart.

Using and you can conclude that which implies thatthe sequence is as follows.


If you know the th term of an arithmetic sequence and you know thecommon difference of the sequence, you can find the th term by using therecursion formula

Recursion formula

With this formula, you can find any term of an arithmetic sequence, provided thatyou know the preceding term. For instance, if you know the first term, you canfind the second term. Then, knowing the second term, you can find the third term,and so on.

Using a Recursion Formula

Find the ninth term of the arithmetic sequence that begins with 2 and 9.

SolutionFor this sequence, the common difference is There are two waysto find the ninth term. One way is simply to write out the first nine terms (byrepeatedly adding 7).

Another way to find the ninth term is to first find a formula for the th term.Because the first term is 2, it follows that

Therefore, a formula for the th term is

which implies that the ninth term is


a9 � 7�9� � 5 � 58.

an � 7n � 5

n

c � a1 � d � 2 � 7 � �5.

n

2, 9, 16, 23, 30, 37, 44, 51, 58

d � 9 � 2 � 7.

an�1 � an � d.

�n � 1�n

a1

5

a2

10

a3

15

a4

20

a5

25

a6

30

a7

35

a8

40

a9

45

a10

50

a11

55

. . .

. . .

d � 5,a13 � 65,a4 � 20

a13a4a13 � a4 � 9d.

da13 � 65.a4 � 20


You can find in Example 3 byusing the alternative recursionform of the th term of an arithmetic sequence, as follows.

5 � a1

20 � a1 � 15

20 � a1 � �4 � 1�5

a4 � a1 � �4 � 1�d

an � a1 � �n � 1�d

n

a1

Example 3

Example 4

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The Sum of a Finite Arithmetic SequenceThere is a simple formula for the sum of a finite arithmetic sequence.

For a proof of the sum of a finite arithmetic sequence, see Proofs in Mathematicson page 723.

Finding the Sum of a Finite Arithmetic Sequence

Find the sum:

SolutionTo begin, notice that the sequence is arithmetic (with a common difference of 2).Moreover, the sequence has 10 terms. So, the sum of the sequence is

Formula for the sum of an arithmetic sequence

Substitute 10 for 1 for and 19 for

Simplify.


Finding the Sum of a Finite Arithmetic Sequence

Find the sum of the integers (a) from 1 to 100 and (b) from 1 to

Solutiona. The integers from 1 to 100 form an arithmetic sequence that has 100 terms.

So, you can use the formula for the sum of an arithmetic sequence, as follows.

Formula for sum of an arithmetic sequence

Substitute 100 for 1 for 100 for

Simplify.

b.

Formula for sum of an arithmetic sequence

Substitute N for 1 for and N for


an.a1,n, �N2

�1 � N�

�n2

�a1 � an�

Sn � 1 � 2 � 3 � 4 � . . . � N

� 5050 � 50�101�

an.a1,n, �100

2�1 � 100�

�n

2�a1 � an�

Sn � 1 � 2 � 3 � 4 � 5 � 6 � . . . � 99 � 100

N.

� 5�20� � 100.

an.a1,n, �10

2�1 � 19�

Sn �n

2�a1 � an�

1 � 3 � 5 � 7 � 9 � 11 � 13 � 15 � 17 � 19.


Note that this formula worksonly for arithmetic sequences.

The Sum of a Finite Arithmetic SequenceThe sum of a finite arithmetic sequence with terms is

Sn �n

2�a1 � an �.

n

Example 5

Example 6

Historical NoteA teacher of Carl FriedrichGauss (1777–1855) asked himto add all the integers from 1to 100. When Gauss returnedwith the correct answer afteronly a few moments, theteacher could only look at himin astounded silence. This iswhat Gauss did:

Sn �100 � 101

2� 5050

Sn

Sn

2Sn

�

�

�

1

100

101

�

�

�

2

99

101

�

�

�

3

98

101

�

�

�

. . .

. . .

. . .

�

�

�

100

1

101

The

Gra

ng

er C

olle

ctio

n

333202_0902.qxd 12/5/05 11:29 AM Page 656


The sum of the first terms of an infinite sequence is the th partial sum.The th partial sum can be found by using the formula for the sum of a finitearithmetic sequence.

Finding a Partial Sum of an Arithmetic Sequence

Find the 150th partial sum of the arithmetic sequence

SolutionFor this arithmetic sequence, and So,

and the th term is Therefore, andthe sum of the first 150 terms is

nth partial sum formula


Simplify.

nth partial sum


Applications

Prize Money

In a golf tournament, the 16 golfers with the lowest scores win cash prizes. Firstplace receives a cash prize of $1000, second place receives $950, third placereceives $900, and so on. What is the total amount of prize money?

SolutionThe cash prizes awarded form an arithmetic sequence in which the commondifference is Because

you can determine that the formula for the th term of the sequence isSo, the 16th term of the sequence is

and the total amount of prize money is



Simplify.


� 8�1250� � $10,000.

a16.a1,n, �16

2�1000 � 250�

S16 �n

2�a1 � a16�

S16 � 1000 � 950 � 900 � . . . � 250

a16 � �50�16� � 1050 � 250,an � �50n � 1050.

n

c � a1 � d � 1000 � ��50� � 1050

d � �50.

� 123,675.

� 75�1649�

a150.a1,n, �150

2�5 � 1644�

S150 �n

2�a1 � a150 �

a150 � 11�150� � 6 � 1644,an � 11n � 6.n

c � a1 � d � 5 � 11 � �6

d � 16 � 5 � 11.a1 � 5

5, 16, 27, 38, 49, . . . .

nnn

Example 7

Example 8

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Total Sales

A small business sells $10,000 worth of skin care products during its first year.The owner of the business has set a goal of increasing annual sales by $7500 eachyear for 9 years. Assuming that this goal is met, find the total sales during thefirst 10 years this business is in operation.

SolutionThe annual sales form an arithmetic sequence in which and

So,

and the th term of the sequence is

This implies that the 10th term of the sequence is

See Figure 9.4.

The sum of the first 10 terms of the sequence is


Substitute 10 for 10,000 for and 77,500 for

Simplify.

Simplify.

So, the total sales for the first 10 years will be $437,500.


� 437,500.

� 5�87,500�

a10.a1,n, �10

2�10,000 � 77,500�

S10 �n

2�a1 � a10�

� 77,500.

a10 � 7500�10� � 2500

an � 7500n � 2500.

n

� 2500

� 10,000 � 7500

c � a1 � d

d � 7500.a1 � 10,000


Example 9

W RITING ABOUT MATHEMATICS

Numerical Relationships Decide whether it is possible to fill in the blanks in eachof the sequences such that the resulting sequence is arithmetic. If so, find arecursion formula for the sequence.

a. , , , , , 11

b. 17, , , , , , , , , 71

c. 2, 6, , , 162

d. 4, 7.5, , , , , , , , , 39

e. 8, 12, , , , 60.75

�7,

Year1 2 3 4 5 6 7 8 9 10

20,000

60,000

80,000

40,000

an = 7500n + 2500

n

anSmall Business

Sale

s (i

n do

llars

)

FIGURE 9.4

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In Exercises 1–10, determine whether the sequence isarithmetic. If so, find the common difference.

1. 2.

3. 4. 80, 40, 20, 10, 5,

5. 6.

7. 1,

8.

9.

10.

In Exercises 11–18, write the first five terms of the sequence.Determine whether the sequence is arithmetic. If so, findthe common difference. (Assume that begins with 1.)

11. 12.

13. 14.

15.

16.

17.

18.

In Exercises 19–30, find a formula for for the arithmeticsequence.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

In Exercises 31–38, write the first five terms of thearithmetic sequence.

31.

32.

33.

34.

35.

36.

37.

38.

In Exercises 39–44, write the first five terms of thearithmetic sequence. Find the common difference andwrite the th term of the sequence as a function of

39.

40.

41.

42.

43.

44.

In Exercises 45–48, the first two terms of the arithmeticsequence are given. Find the missing term.

45.

46.

47.

48. a8 � �a2 � �13.8,a1 � �0.7,

a7 � �a2 � 6.6,a1 � 4.2,

a9 � �a2 � 13,a1 � 3,

a10 � �a2 � 11,a1 � 5,

a1 � 0.375, ak�1 � ak � 0.25

a1 �58, ak�1 � ak �

18

a1 � 72, ak�1 � ak � 6

a1 � 200, ak�1 � ak � 10

a1 � 6, ak�1 � ak � 5

a1 � 15, ak�1 � ak � 4

n.n

a3 � 19, a15 � �1.7

a8 � 26, a12 � 42

a4 � 16, a10 � 46

a1 � 2, a12 � 46

a1 � 16.5, d � 0.25

a1 � �2.6, d � �0.4

a1 � 5, d � �34

a1 � 5, d � 6

a5 � 190, a10 � 115

a3 � 94, a6 � 85

a1 � �4, a5 � 16

a1 � 5, a4 � 15

10, 5, 0, �5, �10, . . .

4, 32, �1, � 72 , . . .

a1 � �y, d � 5y

a1 � x, d � 2x

a1 � 0, d � �23

a1 � 100, d � �8

a1 � 15, d � 4

a1 � 1, d � 3

an

an � �2n �n

an ��1�n3

n

an � 2n�1

an � ��1�n

an � 1 � �n � 1�4an � 3 � 4�n � 2�an � 100 � 3nan � 5 � 3n

n

12, 22, 32, 42, 52, . . .

ln 1, ln 2, ln 3, ln 4, ln 5, . . .

5.3, 5.7, 6.1, 6.5, 6.9, . . .

56, . . .4

3,23,1

3,

3, 52, 2, 32, 1, . . .94, 2, 74, 32, 54, . . .

. . .1, 2, 4, 8, 16, . . .

4, 7, 10, 13, 16, . . .10, 8, 6, 4, 2, . . .

Exercises 9.2


1. A sequence is called an ________ sequence if the differences between two consecutive terms are the same. This difference is called the ________ difference.

2. The th term of an arithmetic sequence has the form ________.

3. The formula can be used to find the sum of the first terms of an arithmetic sequence,

called the ________ of a ________ ________ ________.


nSn �n2

�a1 � an�

n

333202_0902.qxd 12/5/05 11:30 AM Page 659

www.Eduspace.com

In Exercises 49–52, match the arithmetic sequence with itsgraph. [The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

49. 50.

51. 52.

In Exercises 53–56, use a graphing utility to graph the first10 terms of the sequence. (Assume that begins with 1.)

53. 54.

55. 56.

In Exercises 57– 64, find the indicated th partial sum of thearithmetic sequence.

57.

58.

59. 4.2, 3.7, 3.2, 2.7,

60.

61.

62. 75, 70, 65, 60,

63.

64.

65. Find the sum of the first 100 positive odd integers.

66. Find the sum of the integers from to 50.

In Exercises 67–74, find the partial sum.

67. 68.

69. 70.

71. 72.

73. 74.

In Exercises 75–80, use a graphing utility to find the partialsum.

75. 76.

77. 78.

79. 80.

Job Offer In Exercises 81 and 82, consider a job offer withthe given starting salary and the given annual raise.

(a) Determine the salary during the sixth year ofemployment.

(b) Determine the total compensation from the companythrough six full years of employment.

Starting Salary Annual Raise

81. $32,500 $1500

82. $36,800 $1750

83. Seating Capacity Determine the seating capacity of anauditorium with 30 rows of seats if there are 20 seats in thefirst row, 24 seats in the second row, 28 seats in the thirdrow, and so on.

84. Seating Capacity Determine the seating capacity of anauditorium with 36 rows of seats if there are 15 seats in thefirst row, 18 seats in the second row, 21 seats in the thirdrow, and so on.

85. Brick Pattern A brick patio has the approximate shape ofa trapezoid (see figure). The patio has 18 rows of bricks.The first row has 14 bricks and the 18th row has 31 bricks.How many bricks are in the patio?

FIGURE FOR 85 FIGURE FOR 86

86. Brick Pattern A triangular brick wall is made by cuttingsome bricks in half to use in the first column of every otherrow. The wall has 28 rows. The top row is one-half brickwide and the bottom row is 14 bricks wide. How manybricks are used in the finished wall?

14

31

�200

j�1

�4.5 � 0.025j��60

i�1

�250 �83i�

�100

n�0

8 � 3n

16�100

n�1

n � 4

2

�50

n�0

�1000 � 5n��20

n�1

�2n � 5�

�250

n�1

�1000 � n��400

n�1

�2n � 1�

�100

n�51

n � �50

n�1

n�30

n�11

n � �10

n�1

n

�100

n�51

7n�100

n�10

6n

�100

n�1

2n�50

n�1

n

�10

n � 100a1 � 15, a100 � 307,

n � 25a1 � 100, a25 � 220,

n � 25 . . . ,

n � 1040, 37, 34, 31, . . . ,

n � 100.5, 0.9, 1.3, 1.7, . . . ,

n � 12 . . . ,

n � 252, 8, 14, 20, . . . ,

n � 108, 20, 32, 44, . . . ,

n

an � �0.3n � 8an � 0.2n � 3

an � �5 � 2nan � 15 �32n

n

an � 25 � 3nan � 2 �34 n

an � 3n � 5an � �34 n � 8

2 106 84

6

−6

12

18

24

30

n

an

2 106 84

2

−2

4

6

8

10

n

an

84

2

−2−4

4

6

8

10n

an

2 684−6

6

12

18

24

n

an

2 6


333202_0902.qxd 12/5/05 11:30 AM Page 660


87. Falling Object An object with negligible air resistance isdropped from a plane. During the first second of fall, theobject falls 4.9 meters; during the second second, it falls14.7 meters; during the third second, it falls 24.5 meters;during the fourth second, it falls 34.3 meters. If thisarithmetic pattern continues, how many meters will theobject fall in 10 seconds?

88. Falling Object An object with negligible air resistance isdropped from the top of the Sears Tower in Chicago at aheight of 1454 feet. During the first second of fall, theobject falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; during thefourth second, it falls 112 feet. If this arithmetic patterncontinues, how many feet will the object fall in 7 seconds?

89. Prize Money A county fair is holding a baked goodscompetition in which the top eight bakers receive cashprizes. First places receives a cash prize of $200, secondplace receives $175, third place receives $150, and so on.

(a) Write a sequence that represents the cash prizeawarded in terms of the place in which the bakedgood places.

(b) Find the total amount of prize money awarded at thecompetition.

90. Prize Money A city bowling league is holding atournament in which the top 12 bowlers with the highestthree-game totals are awarded cash prizes. First place willwin $1200, second place $1100, third place $1000, and so on.

(a) Write a sequence that represents the cash prizeawarded in terms of the place in which the bowlerfinishes.

(b) Find the total amount of prize money awarded at thetournament.

91. Total Profit A small snowplowing company makes aprofit of $8000 during its first year. The owner of thecompany sets a goal of increasing profit by $1500 eachyear for 5 years. Assuming that this goal is met, find thetotal profit during the first 6 years of this business. Whatkinds of economic factors could prevent the company frommeeting its profit goal? Are there any other factors thatcould prevent the company from meeting its goal? Explain.

92. Total Sales An entrepreneur sells $15,000 worth of sportsmemorabilia during one year and sets a goal of increasingannual sales by $5000 each year for 9 years. Assuming thatthis goal is met, find the total sales during the first 10 yearsof this business. What kinds of economic factors could prevent the business from meeting its goals?

93. Borrowing Money You borrowed $2000 from a friend topurchase a new laptop computer and have agreed to payback the loan with monthly payments of $200 plus 1%interest on the unpaid balance.

(a) Find the first six monthly payments you will make, andthe unpaid balance after each month.

(b) Find the total amount of interest paid over the term ofthe loan.

94. Borrowing Money You borrowed $5000 from yourparents to purchase a used car. The arrangements of theloan are such that you will make payments of $250 permonth plus 1% interest on the unpaid balance.

(a) Find the first year’s monthly payments you will make,and the unpaid balance after each month.

(b) Find the total amount of interest paid over the term ofthe loan.

nan

nan

95. Data Analysis: Personal Income The table showsthe per capita personal income in the United Statesfrom 1993 to 2003. (Source: U.S. Bureau ofEconomic Analysis)

(a) Find an arithmetic sequence that models the data.Let represent the year, with correspondingto 1993.

(b) Use the regression feature of a graphing utility tofind a linear model for the data. How does thismodel compare with the arithmetic sequence youfound in part (a)?

(c) Use a graphing utility to graph the terms of thefinite sequence you found in part (a).

(d) Use the sequence from part (a) to estimate the percapita personal income in 2004 and 2005.

(e) Use your school’s library, the Internet, or someother reference source to find the actual per capitapersonal income in 2004 and 2005, and comparethese values with the estimates from part (d).

n � 3n

an

Model It

Year Per capitapersonal income,

1993 $21,356

1994 $22,176

1995 $23,078

1996 $24,176

1997 $25,334

1998 $26,880

1999 $27,933

2000 $29,848

2001 $30,534

2002 $30,913

2003 $31,633

an

333202_0902.qxd 12/5/05 11:30 AM Page 661

96. Data Analysis: Revenue The table shows the annualrevenue (in millions of dollars) for NextelCommunications, Inc. from 1997 to 2003. (Source:Nextel Communications, Inc.)

(a) Construct a bar graph showing the annual revenuefrom 1997 to 2003.

(b) Use the linear regression feature of a graphing utilityto find an arithmetic sequence that approximates theannual revenue from 1997 to 2003.

(c) Use summation notation to represent the total revenuefrom 1997 to 2003. Find the total revenue.

(d) Use the sequence from part (b) to estimate the annualrevenue in 2008.

Synthesis

True or False? In Exercises 97 and 98, determine whetherthe statement is true or false. Justify your answer.

97. Given an arithmetic sequence for which only the first twoterms are known, it is possible to find the th term.

98. If the only known information about a finite arithmeticsequence is its first term and its last term, then it ispossible to find the sum of the sequence.

99. Writing In your own words, explain what makes asequence arithmetic.

100. Writing Explain how to use the first two terms of anarithmetic sequence to find the th term.

101. Exploration

(a) Graph the first 10 terms of the arithmetic sequence

(b) Graph the equation of the line

(c) Discuss any differences between the graph of

and the graph of

(d) Compare the slope of the line in part (b) with thecommon difference of the sequence in part (a). Whatcan you conclude about the slope of a line and thecommon difference of an arithmetic sequence?

102. Pattern Recognition

(a) Compute the following sums of positive odd integers.

(b) Use the sums in part (a) to make a conjecture about thesums of positive odd integers. Check your conjecturefor the sum

(c) Verify your conjecture algebraically.

103. Think About It The sum of the first 20 terms of anarithmetic sequence with a common difference of 3 is 650.Find the first term.

104. Think About It The sum of the first terms of an arith-metic sequence with first term and common difference

is Determine the sum if each term is increased by 5.Explain.

Skills Review

In Exercises 105–108, find the slope and y-intercept (ifpossible) of the equation of the line. Sketch the line.

105.

106.

107.

108.

In Exercises 109 and 110, use Gauss-Jordan elimination tosolve the system of equations.

109.

110.

111. Make a Decision To work an extended applicationanalyzing the median sales price of existing one-familyhomes in the United States from 1987 to 2003, visit thistext’s website at college.hmco.com. (Data Source:National Association of Realtors)

��x5x8x

�

�

�

4y3y2y

�

�

�

10zz

3z

�

�

�

431

�5

�2x3x6x

�

�

�

y2y5y

�

�

�

7z4zz

�

�

�

�1017

�20

y � 11 � 0

x � 7 � 0

9x � y � �8

2x � 4y � 3

Sn.da1

n

1 � 3 � 5 � 7 � 9 � 11 � 13 � �.

1 � 3 � 5 � 7 � 9 � 11 � �1 � 3 � 5 � 7 � 9 � �1 � 3 � 5 � 7 � �1 � 3 � 5 � �1 � 3 � �

y � 3x � 2.

an � 2 � 3n

y � 3x � 2.

an � 2 � 3n.

n

n

an


Year Revenue,

1997 739

1998 1847

1999 3326

2000 5714

2001 7689

2002 8721

2003 10,820

an

333202_0902.qxd 12/8/05 10:53 AM Page 662

Section 9.3 Geometric Sequences and Series 663

Geometric SequencesIn Section 9.2, you learned that a sequence whose consecutive terms have acommon difference is an arithmetic sequence. In this section, you will studyanother important type of sequence called a geometric sequence. Consecutiveterms of a geometric sequence have a common ratio.

Examples of Geometric Sequences

a. The sequence whose th term is is geometric. For this sequence, thecommon ratio of consecutive terms is 2.

Begin with

b. The sequence whose th term is is geometric. For this sequence, thecommon ratio of consecutive terms is 3.

Begin with

c. The sequence whose th term is is geometric. For this sequence, thecommon ratio of consecutive terms is

Begin with

Now try Exercise 1.

The sequence 1, 4, 9, 16, whose th term is is not geometric. Theratio of the second term to the first term is

but the ratio of the third term to the second term is a3

a2�

94

.

a2

a1�

41

� 4

n2,n. . . ,

1�9�1�3 � �

13

n � 1.�1

3,

1

9, �

1

27,

1

81, . . . , ��

1

3�n

, . . .

�13.

��13�n

n

3612 � 3

n � 1.12, 36, 108, 324, . . . , 4�3n �, . . .

4�3n �n

42 � 2

n � 1.2, 4, 8, 16, . . . , 2n, . . .

2nn

What you should learn• Recognize, write, and find

the nth terms of geometricsequences.

• Find nth partial sums ofgeometric sequences.

• Find the sum of an infinitegeometric series.

• Use geometric sequences to model and solve real-lifeproblems.

Why you should learn itGeometric sequences can beused to model and solve real-life problems. For instance, inExercise 99 on page 670, you willuse a geometric sequence tomodel the population of China.

Geometric Sequences and Series

© Bob Krist /Corbis

9.3

Definition of Geometric SequenceA sequence is geometric if the ratios of consecutive terms are the same. So,the sequence is geometric if there is a number such that

and so the number is the common ratio of the sequence.r

r � 0a2

a1� r, a3

a2� r, a4

a3� r,

ra1, a2, a3, a4, . . . , an . . .

Example 1

333202_0903.qxd 12/5/05 11:32 AM Page 663

In Example 1, notice that each of the geometric sequences has an th termthat is of the form where the common ratio of the sequence is A geometricsequence may be thought of as an exponential function whose domain is the setof natural numbers.

If you know the th term of a geometric sequence, you can find the term by multiplying by That is,

Finding the Terms of a Geometric Sequence

Write the first five terms of the geometric sequence whose first term is andwhose common ratio is Then graph the terms on a set of coordinate axes.

SolutionStarting with 3, repeatedly multiply by 2 to obtain the following.

1st term

2nd term

3rd term

4th term

5th term

Figure 9.5 shows the first five terms of this geometric sequence.


Finding a Term of a Geometric Sequence

Find the 15th term of the geometric sequence whose first term is 20 and whosecommon ratio is 1.05.

SolutionFormula for geometric sequence

Substitute 20 for 1.05 for and 15 for

Use a calculator.


� 39.599

n.r, a1, � 20�1.05�15�1

a15 � a1rn�1

a5 � 3�24� � 48

a4 � 3�23� � 24

a3 � 3�22� � 12

a2 � 3�21� � 6

a1 � 3

r � 2.a1 � 3

an�1 � ran.r.�n � 1�thn

r.arn,n


The nth Term of a Geometric SequenceThe th term of a geometric sequence has the form

where is the common ratio of consecutive terms of the sequence. So, everygeometric sequence can be written in the following form.

. . .a1rn�1,, . . . ,a1r

4a1r3,a1r

2,a1r,a1,

. . . . .an,. . . . . ,a5,a4,a3,a2,a1,

r

an � a1rn�1

n

1 53 42

10

20

30

40

50

n

an

FIGURE 9.5

Example 2

Example 3

333202_0903.qxd 12/5/05 11:32 AM Page 664


Find the 12th term of the geometric sequence

SolutionThe common ratio of this sequence is

Because the first term is you can determine the 12th term to be

Formula for geometric sequence


Use a calculator.

Simplify.


If you know any two terms of a geometric sequence, you can use that infor-mation to find a formula for the th term of the sequence.


The fourth term of a geometric sequence is 125, and the 10th term is Find the 14th term. (Assume that the terms of the sequence are positive.)

SolutionThe 10th term is related to the fourth term by the equation

Multiply 4th term by

Because and you can solve for as follows.

Substitute for and 125 for

Divide each side by 125.

Take the sixth root of each side.

You can obtain the 14th term by multiplying the 10th term by

Multiply the 10th term by

Substitute for and for

Simplify.


�125

1024

r.12a10

12564 �

125

64 �1

2�4

r14�10. a14 � a10r4

r4.

1

2� r

1

64� r 6

a4.a1012564

125

64� 125r6

ra4 � 125,a10 � 125�64

r 10�4.a10 � a4r 6.

125�64.

n

� 885,735.

� 5�177,147�

n.r,a1, a12 � 5�3�12�1

an � a1rn�1

�n � 12�a1 � 5,

r �15

5� 3.

5, 15, 45, . . . .


Remember that is the commonratio of consecutive terms of ageometric sequence. So, inExample 5,

� a4r 6.

� a1 �a2

a1�

a3

a2�

a4

a3� r6

� a1 � r � r � r � r 6

a10 � a1r9

r

Example 4

Example 5

333202_0903.qxd 12/5/05 11:32 AM Page 665

The Sum of a Finite Geometric SequenceThe formula for the sum of a finite geometric sequence is as follows.

For a proof of the sum of a finite geometric sequence, see Proofs in Mathematicson page 723.

Finding the Sum of a Finite Geometric Sequence

Find the sum

SolutionBy writing out a few terms, you have

Now, because and you can apply the formula for thesum of a finite geometric sequence to obtain

Formula for the sum of a sequence

Substitute 4 for 0.3 for and 12 for

Use a calculator.


When using the formula for the sum of a finite geometric sequence, becareful to check that the sum is of the form

Exponent for is

If the sum is not of this form, you must adjust the formula. For instance, if the

sum in Example 6 were then you would evaluate the sum as follows.

a1 � 4�0.3�, r � 0.3, n � 12 � 4�0.3��1 � �0.3�12

1 � 0.3 � � 1.714.

� 4�0.3� � 4�0.3��0.3� � 4�0.3��0.3�2 � . . . � 4�0.3��0.3�11

�12

i�1

4�0.3�i � 4�0.3� � 4�0.3�2 � 4�0.3�3 � . . . � 4�0.3�12

�12

i�1 4�0.3�i,

i � 1.r�n

i�1 a1 r

i�1.

� 5.714.

n.r,a1, �12

i�1

4�0.3�i�1 � 4�1 � �0.3�12

1 � 0.3 �

Sn � a1�1 � rn

1 � r �

n � 12,r � 0.3,a1 � 4,

�12

i�1

4�0.3�i�1 � 4�0.3�0 � 4�0.3�1 � 4�0.3�2 � . . . � 4�0.3�11.

�12

i�1

4�0.3�i�1.


The Sum of a Finite Geometric SequenceThe sum of the finite geometric sequence

with common ratio is given by Sn � �n

i�1

a1 ri�1 � a1�1 � rn

1 � r �.r � 1

a1, a1r, a1r2, a1r

3, a1r4, . . . , a1r

n�1

Example 6

333202_0903.qxd 12/5/05 11:32 AM Page 666

Geometric SeriesThe summation of the terms of an infinite geometric sequence is called aninfinite geometric series or simply a geometric series.

The formula for the sum of a finite geometric sequence can, depending onthe value of be extended to produce a formula for the sum of an infinite geo-metric series. Specifically, if the common ratio has the property that itcan be shown that becomes arbitrarily close to zero as increases withoutbound. Consequently,

as

This result is summarized as follows.

Note that if the series does not have a sum.

Finding the Sum of an Infinite Geometric Series

Find each sum.

a.

b.

Solution

a.

b.


� 3.33

�10

3

a 1

1 � r �

3

1 � 0.1

3 � 0.3 � 0.03 � 0.003 � . . . � 3 � 3�0.1� � 3�0.1�2 � 3�0.1�3 � . . .

� 10

a 1

1 � r �

4

1 � 0.6

��

n�1

4�0.6�n �1 � 4 � 4�0.6� � 4�0.6�2 � 4�0.6�3 � . . . � 4�0.6�n �1 � . . .

3 � 0.3 � 0.03 � 0.003 � . . .

��

n�1

4�0.6�n �1

�r� ≥ 1,

�.na1�1 � 0

1 � r�a1�1 � rn

1 � r �

nrn�r� < 1,r

r,


The Sum of an Infinite Geometric SeriesIf the infinite geometric series

has the sum

S � ��

i�0 a1r

i �a1

1 � r.

a1 � a1r � a1r2 � a1r

3 � . . . � a1rn�1 � . . .

�r� < 1,

Example 7

Use a graphing utility to graph

for and Whathappens as

Use a graphing utility to graph

for 2, and 3. Whathappens as x →�?

r � 1.5,

y � �1 � r x

1 � r �

x →�?

45.2

3,r �12,

y � �1 � r x

1 � r �

Exploration

333202_0903.qxd 12/5/05 11:32 AM Page 667

Application

Increasing Annuity

A deposit of $50 is made on the first day of each month in a savings account thatpays 6% compounded monthly. What is the balance at the end of 2 years? (Thistype of savings plan is called an increasing annuity.)

SolutionThe first deposit will gain interest for 24 months, and its balance will be

The second deposit will gain interest for 23 months, and its balance will be

The last deposit will gain interest for only 1 month, and its balance will be

The total balance in the annuity will be the sum of the balances of the 24deposits. Using the formula for the sum of a finite geometric sequence, with

and you have

Simplify.


� $1277.96.

S24 � 50�1.005��1 � �1.005�24

1 � 1.005 �r � 1.005,A1 � 50�1.005�

� 50�1.005�.

A1 � 50�1 �0.0612 �1

� 50�1.005�23.

A23 � 50�1 �0.0612 �23

� 50�1.005�24.

A24 � 50�1 �0.0612 �24


Example 8

Substitute for 1.005 for and 24 for n.r,

A1,50�1.005�


An Experiment You will need a piece of string or yarn, a pair of scissors, and a tapemeasure. Measure out any length of string at least 5 feet long. Double over thestring and cut it in half. Take one of the resulting halves, double it over, and cut it inhalf. Continue this process until you are no longer able to cut a length of string inhalf. How many cuts were you able to make? Construct a sequence of the resultingstring lengths after each cut, starting with the original length of the string. Find aformula for the nth term of this sequence. How many cuts could you theoreticallymake? Discuss why you were not able to make that many cuts.

Recall from Section 3.1 that theformula for compound interestis

So, in Example 8, $50 is theprincipal 0.06 is the interestrate 12 is the number of compoundings per year and 2 is the time in years. If yousubstitute these values into theformula, you obtain

� 50�1 �0.0612 �24

.

A � 50�1 �0.0612 �12�2�

tn,

r,P,

A � P�1 �rn�

nt

.

333202_0903.qxd 12/5/05 11:32 AM Page 668


In Exercises 1–10, determine whether the sequence isgeometric. If so, find the common ratio.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

In Exercises 11–20, write the first five terms of the geometricsequence.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

In Exercises 21–26, write the first five terms of the geometricsequence. Determine the common ratio and write the thterm of the sequence as a function of

21. 22.

23. 24.

25. 26.

In Exercises 27–34, write an expression for the th term ofthe geometric sequence. Then find the indicated term.

27. 28.

29. 30.

31.

32.

33.

34.

In Exercises 35– 42, find the indicated th term of thegeometric sequence.

35. 9th term: 7, 21, 63, . . .

36. 7th term: 3, 36, 432, . . .

37. 10th term: 5, 30, 180, . . .

38. 22nd term: 4, 8, 16, . . .

39. 3rd term:

40. 1st term:

41. 6th term:

42. 7th term:

In Exercises 43– 46, match the geometric sequence with itsgraph. [The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

43. 44.

45. 46. an � 18��32�n�1

an � 18�32�n�1

an � 18��23�n�1

an � 18�23�n�1

2 108−200−400

−600

200

400

600

n

an

2 108−2

6

12

18

−12

−18

n

an

n2−2 4 6 108

750

600

450

300

150

an

2 106 84

4

−4

8

12

16

20

n

an

a5 �6427a3 �

163 ,

a7 �23a4 � �18,

a5 �364a2 � 3,

a4 �274a1 � 16,

n

a1 � 1000, r � 1.005, n � 60

a1 � 500, r � 1.02, n � 40

a1 � 1, r � 3, n � 8

a1 � 100, r � ex, n � 9

a1 � 64, r � �14, n � 10a1 � 6, r � �

13, n � 12

a1 � 5, r �32, n � 8a1 � 4, r �

12, n � 10

n

a1 � 48, ak�1 � �12 aka1 � 6, ak�1 � �

32ak

a1 � 5, ak�1 � �2aka1 � 7, ak�1 � 2a k

a1 � 81, ak�1 �13aka1 � 64, ak�1 �

12ak

n.n

a1 � 5, r � 2xa1 � 2, r �x

4

a1 � 3, r � 5a1 � 1, r � e

a1 � 6, r � �14a1 � 5, r � �

110

a1 � 1, r �13a1 � 1, r �

12

a1 � 6, r � 2a1 � 2, r � 3

15, 27, 39, 4

11,. . .1, 12, 13, 14,. . .

9, �6, 4, �83,. . .1

8, 14, 12, 1,. . .

5, 1, 0.2, 0.04,. . .1, �12, 14, �1

8,. . .

36, 27, 18, 9,. . .3, 12, 21, 30,. . .

3, 12, 48, 192,. . .5, 15, 45, 135,. . .

Exercises 9.3


1. A sequence is called a ________ sequence if the ratios between consecutive terms are the same. This ratio is called the ________ ratio.

2. The th term of a geometric sequence has the form ________.

3. The formula for the sum of a finite geometric sequence is given by ________.

4. The sum of the terms of an infinite geometric sequence is called a ________ ________.

5. The formula for the sum of an infinite geometric series is given by ________.


n

333202_0903.qxd 12/5/05 11:32 AM Page 669

www.Eduspace.com

In Exercises 47–52, use a graphing utility to graph the first10 terms of the sequence.

47. 48.

49. 50.

51. 52.

In Exercises 53–72, find the sum of the finite geometricsequence.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

71. 72.

In Exercises 73–78, use summation notation to write thesum.

73.

74.

75.

76.

77.

78.

In Exercises 79–92, find the sum of the infinite geometricseries.

79. 80.

81. 82.

83. 84.

85. 86.

87. 88.

89. 90.

91. 92.

In Exercises 93–96, find the rational number representationof the repeating decimal.

93. 94.

95. 96.

Graphical Reasoning In Exercises 97 and 98, use a graph-ing utility to graph the function. Identify the horizontalasymptote of the graph and determine its relationship tothe sum.

97.

98. ��

n �0

2�4

5�n

f �x� � 2�1 � �0.8�x

1 � �0.8� �,

��

n �0

6�1

2�n

f �x� � 6�1 � �0.5�x

1 � �0.5� �,

1.380.318

0.2970.36

�12536 �

256 � 5 � 6 � . . .1

9 �13 � 1 � 3 � . . .

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n�0 �10�0.2�n�

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n�0 2�2

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32 � 24 � 18 � . . . � 10.125

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15 � 3 �35

� . . . �3

625

2 �12

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� . . . �1

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7 � 14 � 28 � . . . � 896

5 � 15 � 45 � . . . � 3645

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n�1

an � 10�1.2�n�1an � 2�1.3�n�1

an � 20��1.25�n�1an � 12��0.4�n�1

an � 10�1.5�n�1an � 12��0.75�n�1


99. Data Analysis: Population The table shows thepopulation of China (in millions) from 1998 through2004. (Source: U.S. Census Bureau)

(a) Use the exponential regression feature of agraphing utility to find a geometric sequence thatmodels the data. Let represent the year, with

corresponding to 1998.

(b) Use the sequence from part (a) to describe the rateat which the population of China is growing.

n � 8n

an

Model It

Year Population,

1998 1250.4

1999 1260.1

2000 1268.9

2001 1276.9

2002 1284.3

2003 1291.5

2004 1298.8

an

333202_0903.qxd 12/5/05 11:32 AM Page 670


100. Compound Interest A principal of $1000 is invested at6% interest. Find the amount after 10 years if the interestis compounded (a) annually, (b) semiannually,(c) quarterly, (d) monthly, and (e) daily.

101. Compound Interest A principal of $2500 is invested at2% interest. Find the amount after 20 years if the interestis compounded (a) annually, (b) semiannually,(c) quarterly, (d) monthly, and (e) daily.

102. Depreciation A tool and die company buys a machinefor $135,000 and it depreciates at a rate of 30% per year.(In other words, at the end of each year the depreciatedvalue is 70% of what it was at the beginning of the year.)Find the depreciated value of the machine after 5 fullyears.

103. Annuities A deposit of $100 is made at the beginning ofeach month in an account that pays 6%, compoundedmonthly. The balance in the account at the end of 5 years is

Find

104. Annuities A deposit of $50 is made at the beginning ofeach month in an account that pays 8%, compoundedmonthly. The balance in the account at the end of 5 years is

Find

105. Annuities A deposit of dollars is made at thebeginning of each month in an account earning an annualinterest rate compounded monthly. The balance after

years is

Show that the balance is

106. Annuities A deposit of dollars is made at thebeginning of each month in an account earning an annualinterest rate compounded continuously. The balance after years is Show that the balance is

Annuities In Exercises 107–110, consider making monthlydeposits of dollars in a savings account earning an annualinterest rate Use the results of Exercises 105 and 106 to find the balance after years if the interest iscompounded (a) monthly and (b) continuously.

107.108.109.110.

111. Annuities Consider an initial deposit of dollars in anaccount earning an annual interest rate compoundedmonthly. At the end of each month, a withdrawal of dollars will occur and the account will be depleted in years. The amount of the initial deposit required is

Show that the initial deposit is

112. Annuities Determine the amount required in a retire-ment account for an individual who retires at age 65 andwants an income of $2000 from the account each monthfor 20 years. Use the result of Exercise 111 and assumethat the account earns 9% compounded monthly.

Multiplier Effect In Exercises 113–116, use the followinginformation. A tax rebate has been given to propertyowners by the state government with the anticipation thateach property owner spends approximately of therebate, and in turn each recipient of this amount spends

of what they receive, and so on. Economists refer to thisexchange of money and its circulation within the economyas the “multiplier effect.” The multiplier effect operates onthe idea that the expenditures of one individual becomethe income of another individual. For the given tax rebate,find the total amount put back into the state’s economy, ifthis effect continues without end.

Tax rebate

113. $400 75%114. $250 80%115. $600 72.5%116. $450 77.5%

p%

p%

p%

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r ��1 � �1 �r

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W�1 �r

12��12t

.

P � W�1 �r

12��1

� W�1 �r

12��2

� . . . �

tW

r,P

P � $20, r � 6%, t � 50 yearsP � $100, r � 10%, t � 40 yearsP � $75, r � 3%, t � 25 yearsP � $50, r � 7%, t � 20 years

tAr.

P

A �Per�12�er t � 1�

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P

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A

A.

A � 100�1 �0.06

12 �1

� . . . � 100�1 �0.06

12 �60

.

A


(c) Use the sequence from part (a) to predict the popu-lation of China in 2010. The U.S. Census Bureaupredicts the population of China will be 1374.6million in 2010. How does this value compare withyour prediction?

(d) Use the sequence from part (a) to determine whenthe population of China will reach 1.32 billion.

333202_0903.qxd 12/5/05 11:32 AM Page 671

117. Geometry The sides of a square are 16 inches in length.A new square is formed by connecting the midpoints ofthe sides of the original square, and two of the resultingtriangles are shaded (see figure). If this process is repeatedfive more times, determine the total area of the shadedregion.

118. Sales The annual sales (in millions of dollars) forUrban Outfitters for 1994 through 2003 can be approxi-mated by the model

4, 5, 13

where represents the year, with corresponding to1994. Use this model and the formula for the sum of afinite geometric sequence to approximate the total salesearned during this 10-year period. (Source: UrbanOutfitters Inc.)

119. Salary An investment firm has a job opening with asalary of $30,000 for the first year. Suppose that duringthe next 39 years, there is a 5% raise each year. Find thetotal compensation over the 40-year period.

120. Distance A ball is dropped from a height of 16 feet.Each time it drops feet, it rebounds feet.

(a) Find the total vertical distance traveled by the ball.

(b) The ball takes the following times (in seconds) foreach fall.

Beginning with the ball takes the same amount oftime to bounce up as it does to fall, and so the totaltime elapsed before it comes to rest is

Find this total time.

Synthesis

True or False? In Exercises 121 and 122, determinewhether the statement is true or false. Justify your answer.

121. A sequence is geometric if the ratios of consecutivedifferences of consecutive terms are the same.

122. You can find the th term of a geometric sequence bymultiplying its common ratio by the first term of thesequence raised to the th power.

123. Writing Write a brief paragraph explaining why theterms of a geometric sequence decrease in magnitudewhen

124. Find two different geometric series with sums of 4.

Skills Review

In Exercises 125–128, evaluate the function forand

125.

126.

127.

128.

In Exercises 129–132, completely factor the expression.

129.

130.

131.

132.

In Exercises 133–138, perform the indicated operation(s)and simplify.

133.

134.

135.

136.

137.

138.

139. Make a Decision To work an extended applicationanalyzing the amounts spent on research and developmentin the United States from 1980 to 2003, visit this text’swebsite at college.hmco.com. (Data Source: U.S.Census Bureau)

8 �x � 1x � 4

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x � 4�x � 1��x � 4�

5 �7

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x � 5x � 3

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x3

�3x

6x � 3

x � 2x � 7

�2x�x � 7�6x�x � 2�

3x � 3

�x�x � 3�

x � 3

16x2 � 4x 4

6x2 � 13x � 5

x2 � 4x � 63

9x3 � 64x

g� f �x � 1��f �g�x � 1��f �x � 1�g�x � 1�

g�x� � x 2 � 1.f �x� � 3x � 1

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333202_0903.qxd 12/5/05 11:32 AM Page 672

Section 9.4 Mathematical Induction 673

IntroductionIn this section, you will study a form of mathematical proof called mathematicalinduction. It is important that you see clearly the logical need for it, so take acloser look at the problem discussed in Example 5 in Section 9.2.

Judging from the pattern formed by these first five sums, it appears that the sumof the first odd integers is

Although this particular formula is valid, it is important for you to see thatrecognizing a pattern and then simply jumping to the conclusion that the patternmust be true for all values of is not a logically valid method of proof. There aremany examples in which a pattern appears to be developing for small values of

and then at some point the pattern fails. One of the most famous cases of thiswas the conjecture by the French mathematician Pierre de Fermat (1601–1665),who speculated that all numbers of the form

are prime. For and 4, the conjecture is true.

The size of the next Fermat number is so great that it wasdifficult for Fermat to determine whether it was prime or not. However, anotherwell-known mathematician, Leonhard Euler (1707–1783), later found thefactorization

which proved that is not prime and therefore Fermat’s conjecture was false.Just because a rule, pattern, or formula seems to work for several values of

you cannot simply decide that it is valid for all values of without goingthrough a legitimate proof. Mathematical induction is one method of proof.

nn,

F5

� 641�6,700,417�

F5 � 4,294,967,297

�F5 � 4,294,967,297�

F4 � 65,537

F3 � 257

F2 � 17

F1 � 5

F0 � 3

n � 0, 1, 2, 3,

Fn � 22n� 1, n � 0, 1, 2, . . .

n

n

Sn � 1 � 3 � 5 � 7 � 9 � . . . � �2n � 1� � n2.

n

S5 � 1 � 3 � 5 � 7 � 9 � 52

S4 � 1 � 3 � 5 � 7 � 42

S3 � 1 � 3 � 5 � 32

S2 � 1 � 3 � 22

S1 � 1 � 12

What you should learn• Use mathematical induction to

prove statements involving apositive integer

• Recognize patterns and writethe th term of a sequence.

• Find the sums of powers ofintegers.

• Find finite differences ofsequences.

Why you should learn itFinite differences can be used to determine what type of model can be used to represent a sequence. For instance, inExercise 61 on page 682, you will use finite differences to finda model that represents thenumber of individual income taxreturns filed in the United Statesfrom 1998 to 2003.

n

n.

Mathematical Induction

Mario Tama/Getty Images

9.4

333202_0904.qxd 12/5/05 11:35 AM Page 673

To apply the Principle of Mathematical Induction, you need to be able todetermine the statement for a given statement To determine substitute the quantity for in the statement

A Preliminary Example

Find the statement for each given statement

a.

b.

c.

d.

Solution

a. Replace by

Simplify.

b.

c.

d.

Now try Exercise 1.

A well-known illustration used to explain why the Principle of MathematicalInduction works is the unending line of dominoes shown in Figure 9.6. If the lineactually contains infinitely many dominoes, it is clear that you could not knockthe entire line down by knocking down only one domino at a time. However,suppose it were true that each domino would knock down the next one as it fell.Then you could knock them all down simply by pushing the first one and startinga chain reaction. Mathematical induction works in the same way. If the truth of

implies the truth of and if is true, the chain reaction proceeds asfollows: implies implies implies and so on.P4,P3, P3P2, P2P1

P1Pk�1Pk

3k�1 ≥ 2k � 3

Pk�1 : 3k�1 ≥ 2�k � 1� � 1

k � 4 < 5�k2 � 2k � 1�

Pk�1: �k � 1� � 3 < 5�k � 1�2

� 1 � 5 � 9 � . . . � �4k � 3� � �4k � 1�Pk�1 : Sk�1 � 1 � 5 � 9 � . . . � �4��k � 1� � 1� � 3� � �4�k � 1� � 3�

��k � 1�2�k � 2�2

4

k � 1.kPk�1 : Sk�1 ��k � 1�2�k � 1 � 1�2

4

Pk : 3k ≥ 2k � 1

Pk : k � 3 < 5k2

Pk : Sk � 1 � 5 � 9 � . . . � �4�k � 1� � 3� � �4k � 3�

Pk : Sk �k2�k � 1�2

4

Pk.Pk�1

Pk.kk � 1Pk�1,Pk.Pk�1


It is important to recognize thatin order to prove a statement by induction, both parts of thePrinciple of MathematicalInduction are necessary.

The Principle of Mathematical InductionLet be a statement involving the positive integer If

1. is true, and

2. for every positive integer the truth of implies the truth of

then the statement must be true for all positive integers n.Pn

Pk�1Pkk,

P1

n.Pn

Example 1

FIGURE 9.6

333202_0904.qxd 12/5/05 11:35 AM Page 674

When using mathematical induction to prove a summation formula (such asthe one in Example 2), it is helpful to think of as

where is the th term of the original sum.

Using Mathematical Induction

Use mathematical induction to prove the following formula.

SolutionMathematical induction consists of two distinct parts. First, you must show thatthe formula is true when

1. When the formula is valid, because

The second part of mathematical induction has two steps. The first step is toassume that the formula is valid for some integer The second step is to use thisassumption to prove that the formula is valid for the next integer,

2. Assuming that the formula

is true, you must show that the formula is true.

Group terms to form

Replace by

Combining the results of parts (1) and (2), you can conclude by mathematicalinduction that the formula is valid for all positive integer values of

Now try Exercise 5.

It occasionally happens that a statement involving natural numbers is nottrue for the first positive integers but is true for all values of In theseinstances, you use a slight variation of the Principle of Mathematical Inductionin which you verify rather than This variation is called the extendedprinciple of mathematical induction. To see the validity of this, note from Figure9.6 that all but the first dominoes can be knocked down by knocking overthe th domino. This suggests that you can prove a statement to be true for

by showing that is true and that implies In Exercises 17–22 ofthis section, you are asked to apply this extension of mathematical induction.

Pk�1.PkPkn ≥ kPnk

k � 1

P1.Pk

n ≥ k.k � 1

n.

� �k � 1�2

k 2.Sk � k2 � 2k � 1

Sk. � Sk � �2k � 1�

� �1 � 3 � 5 � 7 � . . . � �2k � 1�� 2k � 2 � 1�

Sk�1 � 1 � 3 � 5 � 7 � . . . � �2k � 1� � �2�k � 1� � 1�

Sk�1 � �k � 1�2

� k2

Sk � 1 � 3 � 5 � 7 � . . . � �2k � 1�

k � 1.k.

S1 � 1 � 12.

n � 1,

n � 1.

� n2

Sn � 1 � 3 � 5 � 7 � . . . � �2n � 1�

�k � 1�ak�1

Sk�1 � Sk � ak�1

Sk�1


Example 2

333202_0904.qxd 12/5/05 11:35 AM Page 675

Using Mathematical Induction

Use mathematical induction to prove the formula

for all integers

Solution1. When the formula is valid, because

2. Assuming that

you must show that

To do this, write the following.

Substitute for

By assumption

Combine fractions.

Factor.

Simplify.

implies

Combining the results of parts (1) and (2), you can conclude by mathematicalinduction that the formula is valid for all integers


When proving a formula using mathematical induction, the only statementthat you need to verify is As a check, however, it is a good idea to tryverifying some of the other statements. For instance, in Example 3, try verifying

and P3.P2

P1.

n ≥ 1.

Sk�1.Sk ��k � 1��k � 2��2k � 3�

6

��k � 1��2k2 � 7k � 6�

6

��k � 1��k�2k � 1� � 6�k � 1��

6

�k�k � 1��2k � 1� � 6�k � 1�2

6

�k�k � 1��2k � 1�

6� �k � 1�2

Sk. � �12 � 22 � 32 � 42 � . . . � k2� � �k � 1�2

Sk�1 � Sk � ak�1

��k � 1��k � 2��2k � 3�

6 .

Sk�1 ��k � 1��k � 1 � 1��2�k � 1� � 1�

6

�k�k � 1��2k � 1�

6

ak � k2Sk � 12 � 22 � 32 � 42 � . . . � k2

S1 � 12 �1�2��3�

6 .

n � 1,

n ≥ 1.

�n�n � 1��2n � 1�

6 Sn � 12 � 22 � 32 � 42 � . . . � n2


Remember that when addingrational expressions, you mustfirst find the least commondenominator (LCD). In Example 3, the LCD is 6.

Example 3

333202_0904.qxd 12/5/05 11:35 AM Page 676

Proving an Inequality by Mathematical Induction

Prove that for all positive integers

Solution1. For and the statement is true because

and

2. Assuming that

you need to show that For you have

By assumption

Because for all it follows that

or

Combining the results of parts (1) and (2), you can conclude by mathematicalinduction that for all integers


Proving Factors by Mathematical Induction

Prove that 3 is a factor of for all positive integers

Solution1. For the statement is true because

So, 3 is a factor.

2. Assuming that 3 is a factor of you must show that 3 is a factor ofTo do this, write the following.

Subtract and add

Regroup terms.

Simplify.

Because 3 is a factor of and 3 is also a factor of it follows that 3 isa factor of Combining the results of parts (1) and (2), you canconclude by mathematical induction that 3 is a factor of for all positiveintegers


Pattern RecognitionAlthough choosing a formula on the basis of a few observations does notguarantee the validity of the formula, pattern recognition is important. Once youhave a pattern or formula that you think works, you can try using mathematicalinduction to prove your formula.

n.4n � 1

4k�1 � 1.4k � 1,4k � 3

� 4k � 3 � �4k � 1�

� 4k�4 � 1� � �4k � 1�

4k. 4k�1 � 1 � 4k�1 � 4k � 4k � 1

4k�1 � 1.4k � 1,

41 � 1 � 3.

n � 1,

n.4n � 1

n ≥ 1.n < 2n

k � 1 < 2k�1.2k�1 > 2k > k � 1

k > 1,2k � k � k > k � 1

2k�1 � 2�2k � > 2�k� � 2k.

n � k,k � 1 < 2k�1.

k < 2k

2 < 22.1 < 21

n � 2,n � 1

n.n < 2n


Example 4

Example 5

To check a result that you have proved by mathematicalinduction, it helps to list thestatement for several values of For instance, in Example 4, youcould list

From this list, your intuitionconfirms that the statement

is reasonable.n < 2n

6 < 26 � 64,5 < 25 � 32,

4 < 24 � 16,2 < 23 � 8,

2 < 22 � 4,1 < 21 � 2,

n.

333202_0904.qxd 12/5/05 11:35 AM Page 677

Finding a Formula for a Finite Sum

Find a formula for the finite sum and prove its validity.

SolutionBegin by writing out the first few sums.

From this sequence, it appears that the formula for the th sum is

To prove the validity of this hypothesis, use mathematical induction. Note thatyou have already verified the formula for so you can begin by assumingthat the formula is valid for and trying to show that it is valid for

By assumption

So, by mathematical induction, you can conclude that the hypothesis is valid.


�k � 1

k � 2�

�k � 1�2

�k � 1��k � 2��

k2 � 2k � 1

�k � 1��k � 2� �

k�k � 2� � 1

�k � 1��k � 2�

�k

k � 1�

1

�k � 1��k � 2�

Sk�1 � � 1

1 � 2�

1

2 � 3�

1

3 � 4�

1

4 � 5� . . . �

1

k�k � 1��1

�k � 1��k � 2�

n � k � 1.n � k

n � 1,

Sk �1

1 � 2�

1

2 � 3�

1

3 � 4�

1

4 � 5� . . . �

1

k�k � 1��

k

k � 1.

k

S4 �1

1 � 2�

1

2 � 3�

1

3 � 4�

1

4 � 5�

48

60�

4

5�

4

4 � 1

S3 �1

1 � 2�

1

2 � 3�

1

3 � 4�

9

12�

3

4�

3

3 � 1

S2 �1

1 � 2�

1

2 � 3�

4

6�

2

3�

2

2 � 1

S1 �1

1 � 2�

1

2�

1

1 � 1

1

1 � 2�

1

2 � 3�

1

3 � 4�

1

4 � 5� . . . �

1

n�n � 1�


Finding a Formula for the nth Term of a SequenceTo find a formula for the th term of a sequence, consider these guidelines.

1. Calculate the first several terms of the sequence. It is often a good idea towrite the terms in both simplified and factored forms.

2. Try to find a recognizable pattern for the terms and write a formula forthe th term of the sequence. This is your hypothesis or conjecture. Youmight try computing one or two more terms in the sequence to test yourhypothesis.

3. Use mathematical induction to prove your hypothesis.

n

n

Example 6

333202_0904.qxd 12/5/05 11:35 AM Page 678

Sums of Powers of IntegersThe formula in Example 3 is one of a collection of useful summation formulas.This and other formulas dealing with the sums of various powers of the first positive integers are as follows.

Finding a Sum of Powers of Integers

Find each sum.

a. b.

Solutiona. Using the formula for the sum of the cubes of the first positive integers, you

obtain

Formula 3

b.

Formula 1 and 2


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� 6�10� � 4�30�

� 6�4�4 � 1�2 � � 4�4�4 � 1��8 � 1�

6 �

� 64

i�1i � 4

4

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4

i�1�6i � 4i2� �

4

i�16i �

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i�14i2

� 784.�49�64�

4 �

72�7 � 1�2

4

7

i�1

i3 � 13 � 23 � 33 � 43 � 53 � 63 � 73

n

4

i�1�6i � 4i2�

7

i�1

i3 � 13 � 23 � 33 � 43 � 53 � 63 � 73

n


Sums of Powers of Integers

1.

2.

3.

4.

5. 15 � 25 � 35 � 45 � . . . � n5 �n2�n � 1�2�2n2 � 2n � 1�

12

14 � 24 � 34 � 44 � . . . � n4 �n�n � 1��2n � 1��3n2 � 3n � 1�

30

13 � 23 � 33 � 43 � . . . � n3 �n2�n � 1�2

4

12 � 22 � 32 � 42 � . . . � n2 �n�n � 1��2n � 1�

6

1 � 2 � 3 � 4 � . . . � n �n�n � 1�

2

Example 7

333202_0904.qxd 12/5/05 11:35 AM Page 679

Finite DifferencesThe first differences of a sequence are found by subtracting consecutive terms.The second differences are found by subtracting consecutive first differences.The first and second differences of the sequence are asfollows.

For this sequence, the second differences are all the same. When this happens,the sequence has a perfect quadratic model. If the first differences are all thesame, the sequence has a linear model. That is, it is arithmetic.

Finding a Quadratic Model

Find the quadratic model for the sequence

SolutionYou know from the second differences shown above that the model is quadraticand has the form

By substituting 1, 2, and 3 for you can obtain a system of three linear equa-tions in three variables.

Substitute 1 for

Substitute 2 for

Substitute 3 for

You now have a system of three equations in and

Using the techniques discussed in Chapter 7, you can find the solution tobe and So, the quadratic model is

Try checking the values of and


a3.a1, a2,

an �1

2n2 �

1

2n � 2.

c � 2.b �12,a �

12,

Equation 1

Equation 2

Equation 3

a � b � c � 3

4a � 2b � c � 5

9a � 3b � c � 8

c.b,a,

n. a3 � a�3�2 � b�3� � c � 8

n. a2 � a�2�2 � b�2� � c � 5

n. a1 � a�1�2 � b�1� � c � 3

n,

an � an2 � bn � c.

3, 5, 8, 12, 17, 23, . . . .

3, 5, 8, 12, 17, 23, . . .


n: 1 2 3 4 5 6

3 5 8 12 17 23

First differences: 2 3 4 5 6

Second differences: 1 1 1 1

an:

Example 8

For a linear model, the firstdifferences should be the samenonzero number. For a quadraticmodel, the second differencesare the same nonzero number.

333202_0904.qxd 12/5/05 11:35 AM Page 680


In Exercises 1–4, find for the given

1. 2.

3. 4.

In Exercises 5–16, use mathematical induction to prove theformula for every positive integer

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

In Exercises 17–22, prove the inequality for the indicatedinteger values of

17. 18.

19.

20. and

21. and

22.

In Exercises 23–34, use mathematical induction to provethe property for all positive integers

23. 24.

25. If then

26. If then

27. Generalized Distributive Law:

28. and are complex conjugates for all

29. A factor of is 3.






In Exercises 35– 40, find a formula for the sum of the first terms of the sequence.

35. 36.

37. 38.

39.

40.1

�n � 1��n � 2�, . . .

1

2 � 3,

1

3 � 4,

1

4 � 5,

1

5 � 6, . . . ,

1

4,

1

12,

1

24,

1

40, . . . ,

1

2n�n � 1�, . . .

3, �92, 27

4 , �818 , . . .1, 9

10, 81100, 729

1000, . . .

25, 22, 19, 16, . . .1, 5, 9, 13, . . .

n

�22n�1 � 32n�1��24n�2 � 1��22n�1 � 1��n4 � n � 4��n3 � n � 3��n3 � 3n2 � 2n�

n ≥ 1.�a � bi�n�a � bi�n

x�y1 � y2 � . . . � yn � � xy1 � xy2 � . . . � xyn

� . . . � ln xn .ln�x1x2 . . . xn � � ln

x1 � ln x2

x1 > 0, x2 > 0, . . . , xn > 0,

�x1 x2 x3 . . . xn ��1 � x1

�1x2�1x3

�1 . . . xn�1.

x1 � 0, x2 � 0, . . . , xn � 0,

�a

b�n

�an

bn�ab�n � an bn

n.

n ≥ 32n2 > �n � 1�2,

a > 0n ≥ 1�1 � a�n ≥ na,

0 < x < yn ≥ 1�x

y�n�1

< �x

y�n

,

1 1

�1 2

�1 3

� . . . �1 n

> n, n ≥ 2

�43�n

> n, n ≥ 7n! > 2n, n ≥ 4

n.

n

i�1

1

�2i � 1��2i � 1��

n

2n � 1

n

i�1

i�i � 1� �n�n � 1��n � 2�

3

n

i�1

i4 �n�n � 1��2n � 1��3n2 � 3n � 1�

30

n

i�1

i5 �n2�n � 1�2�2n2 � 2n � 1�

12

13 � 23 � 33 � 43 � . . . � n3 �n2�n � 1�2

4

1 � 2 � 3 � 4 � . . . � n �n�n � 1�

2

2�1 � 3 � 32 � 33 � . . . � 3n�1� � 3n � 1

1 � 2 � 22 � 23 � . . . � 2n�1 � 2n � 1

1 � 4 � 7 � 10 � . . . � �3n � 2� �n

2�3n � 1�

2 � 7 � 12 � 17 � . . . � �5n � 3� �n

2�5n � 1�

3 � 7 � 11 � 15 � . . . � �4n � 1� � n�2n � 1�2 � 4 � 6 � 8 � . . . � 2n � n�n � 1�

n.

Pk �k

3�2k � 1�Pk �

k2�k � 1�2

4

Pk �1

2�k � 2�Pk �5

k�k � 1�

Pk.Pk�1

Exercises 9.4


1. The first step in proving a formula by ________ ________ is to show that the formula is true when

2. The ________ differences of a sequence are found by subtracting consecutive terms.

3. A sequence is an ________ sequence if the first differences are all the same nonzero number.

4. If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model.


n � 1.

333202_0904.qxd 12/5/05 11:35 AM Page 681

www.Eduspace.com

In Exercises 41–50, find the sum using the formulas for thesums of powers of integers.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

In Exercises 51–56, write the first six terms of the sequencebeginning with the given term. Then calculate the first andsecond differences of the sequence. State whether thesequence has a linear model, a quadratic model, or neither.

51. 52.

53. 54.

55. 56.

In Exercises 57–60, find a quadratic model for the sequencewith the indicated terms.

57.

58.

59.

60.

Synthesis

62. Writing In your own words, explain what is meant by aproof by mathematical induction.

True or False? In Exercises 63–66, determine whether thestatement is true or false. Justify your answer.

63. If the statement is true but the true statement does notimply that the statement is true, then is not necessarilytrue for all positive integers

64. If the statement is true and implies then isalso true.

65. If the second differences of a sequence are all zero, thenthe sequence is arithmetic.

66. A sequence with terms has second differences.

Skills Review

In Exercises 67–70, find the product.

67. 68.

69. 70.

In Exercises 71–74, (a) state the domain of the function,(b) identify all intercepts, (c) find any vertical and horizontalasymptotes, and (d) plot additional solution points asneeded to sketch the graph of the rational function.

71.

72.

73.

74. f �x� �5 � x1 � x

h�t� �t � 7

t

g�x� �x2

x2 � 4

f �x� �x

x � 3

�2x � 4y�3�5 � 4x�3

�2x � y�2�2x2 � 1�2

n � 1n

P1Pk�1,PkPk

n.PnP7

P6P1

a0 � 3, a2 � 0, a6 � 36

a0 � �3, a2 � 1, a4 � 9

a0 � 7, a1 � 6, a3 � 10

a0 � 3, a1 � 3, a4 � 15

an � an�1 � nan � �an�1�2

a0 � 0a0 � 2

an � �2an�1an � an�1 � n

a2 � �3a1 � 3

an � an�1 � 2an � an�1 � 3

a1 � 2a1 � 0

10

j�1

�3 �12 j �

12 j2�

6

i�1

�6i � 8i3�

20

n�1

�n3 � n�6

n�1

�n2 � n�

8

n�1

n55

n�1

n4

10

n�1

n36

n�1

n2

30

n�1

n15

n�1

n


61. Data Analysis: Tax Returns The table shows thenumber (in millions) of individual tax returns filed inthe United States from 1998 to 2003. (Source:Internal Revenue Service)

an

Model It

Year Number of returns,

1998 120.3

1999 122.5

2000 124.9

2001 127.1

2002 129.4

2003 130.3

an


(a) Find the first differences of the data shown in thetable.

(b) Use your results from part (a) to determine whethera linear model can be used to approximate the data.If so, find a model algebraically. Let represent theyear, with corresponding to 1998.

(c) Use the regression feature of a graphing utility tofind a linear model for the data. Compare thismodel with the one from part (b).

(d) Use the models found in parts (b) and (c) toestimate the number of individual tax returns filedin 2008. How do these values compare?

n � 8n

333202_0904.qxd 12/5/05 11:35 AM Page 682

Section 9.5 The Binomial Theorem 683

What you should learn• Use the Binomial Theorem to

calculate binomial coefficients.

• Use Pascal’s Triangle to calcu-late binomial coefficients.

• Use binomial coefficients towrite binomial expansions.

Why you should learn itYou can use binomial coefficientsto model and solve real-lifeproblems. For instance, inExercise 80 on page 690, you willuse binomial coefficients to writethe expansion of a model thatrepresents the amounts of childsupport collected in the U. S.

The Binomial Theorem

© Vince Streano/Corbis

9.5

Binomial CoefficientsRecall that a binomial is a polynomial that has two terms. In this section, you willstudy a formula that gives a quick method of raising a binomial to a power. Tobegin, look at the expansion of for several values of

There are several observations you can make about these expansions.

1. In each expansion, there are terms.

2. In each expansion, and have symmetrical roles. The powers of decreaseby 1 in successive terms, whereas the powers of increase by 1.

3. The sum of the powers of each term is For instance, in the expansion ofthe sum of the powers of each term is 5.

4. The coefficients increase and then decrease in a symmetric pattern.

The coefficients of a binomial expansion are called binomial coefficients. Tofind them, you can use the Binomial Theorem.

For a proof of the Binomial Theorem, see Proofs in Mathematics on page 724.

�x � y�5 � x5 � 5x4y1 � 10x3y2 � 10x2y3 � 5x1y4 � y5

3 � 2 � 54 � 1 � 5

�x � y�5,n.

yxyx

n � 1

�x � y�5 � x5 � 5x4y � 10x3y2 � 10x2y3 � 5xy4 � y5

�x � y�4 � x4 � 4x3y � 6x2y2 � 4xy3 � y4

�x � y�3 � x3 � 3x2y � 3xy2 � y3

�x � y�2 � x 2 � 2xy � y 2

�x � y�1 � x � y

�x � y�0 � 1

n.�x � y�n

The Binomial TheoremIn the expansion of

the coefficient of is

The symbol is often used in place of to denote binomial coefficients.nCr�n

r�

nCr �n!

�n � r�!r!.

xn�r y r

�x � y�n � xn � nxn�1y � . . . �nCr xn�1yr � . . . � nxyn�1 � yn

�x � y�n

333202_0905.qxd 12/5/05 11:37 AM Page 683


Most graphing calculators are programmed to evaluate Consult the user’s guide for yourcalculator and then evaluate You should get an answer of 56.

8C5.

nC r .

Techno logy Finding Binomial Coefficients

Find each binomial coefficient.

a. b. c. d.

Solution

a.

b.

c. d.

Now try Exercise 1.

When and as in parts (a) and (b) above, there is a simple patternfor evaluating binomial coefficients that works because there will always befactorial terms that divide out from the expression.

and

Finding Binomial Coefficients

Find each binomial coefficient.

a. b. c. d.

Solution

a.

b.

c.

d.

Now try Exercise 7.

It is not a coincidence that the results in parts (a) and (b) of Example 2 are thesame and that the results in parts (c) and (d) are the same. In general, it is true that

This shows the symmetric property of binomial coefficients that was identifiedearlier.

nCr � nCn�r.

�12

11� �12!

1! � 11!�

�12� � 11!

1! � 11!�

12

1� 12

12C1 �12

1� 12

�7

4� �7 � 6 � 5 � 4

4 � 3 � 2 � 1� 35

7C3 �7 � 6 � 5

3 � 2 � 1� 35

�1211�12C1�7

4�7C3

3 factors2 factors

�10

3 � �10 � 9 � 8

3 � 2 � 18C2 �8 � 7

2 � 1

3 factors2 factors

r � n,r � 0

�8

8� �8!

0! � 8!� 17C0 �

7!

7! � 0!� 1

�10

3 � �10!

7! � 3!�

�10 � 9 � 8� � 7!

7! � 3!�

10 � 9 � 8

3 � 2 � 1� 120

8C2 �8!

6! � 2!�

�8 � 7� � 6!

6! � 2!�

8 � 7

2 � 1� 28

�88�7C0�10

3 �8C2

Example 1

Example 2

333202_0905.qxd 12/5/05 11:37 AM Page 684


Pascal’s TriangleThere is a convenient way to remember the pattern for binomial coefficients. Byarranging the coefficients in a triangular pattern, you obtain the following array,which is called Pascal’s Triangle. This triangle is named after the famous Frenchmathematician Blaise Pascal (1623–1662).

The first and last numbers in each row of Pascal’s Triangle are 1. Every othernumber in each row is formed by adding the two numbers immediately above thenumber. Pascal noticed that numbers in this triangle are precisely the samenumbers that are the coefficients of binomial expansions, as follows.

The top row in Pascal’s Triangle is called the zeroth row because it corresponds to the binomial expansion Similarly, the next row iscalled the first row because it corresponds to the binomial expansion

In general, the nth row in Pascal’s Triangle gives thecoefficients of

Using Pascal’s Triangle

Use the seventh row of Pascal’s Triangle to find the binomial coefficients.

Solution


8C88C78C68C58C48C38C28C18C0

8C0, 8C1, 8C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8

�x � y�n .�x � y�1 � 1�x� � 1�y�.

�x � y�0 � 1.

�x � y�7 � 1x7 � 7x6y � 21x5y2 � 35x4y3 � 35x3y4 � 21x2y5 � 7xy6 � 1y7

�x � y�6 � 1x6 � 6x5y � 15x4y2 � 20x3y3 � 15x2y4 � 6xy5 � 1y6

�x � y�5 � 1x5 � 5x4y � 10x3y2 � 10x2y3 � 5xy4 � 1y5

�x � y�4 � 1x4 � 4x3y � 6x2y2 � 4xy3 � 1y4

�x � y�3 � 1x3 � 3x2y � 3xy2 � 1y3

�x � y�2 � 1x2 � 2xy � 1y2

�x � y�1 � 1x � 1y

�x � y�0 � 1

11

1

7

1

6

1

5

21

1

4

15

1

3

10

35

1

2

6

20

1

3

10

35

1

4

15

1

5

21

1

61

71

1 15 � 6 � 21

4 � 6 � 10

0th row

1st row

2nd row

3rd row

�

1

1

8

7

28

21

56

35

70

35

56

21

28

7

8

1

1

Complete the table and describethe result.

What characteristic of Pascal’sTriangle is illustrated by thistable?

Exploration

n r

9 5

7 1

12 4

6 0

10 7 ��

��

��

��

��nCn�rnCr

Example 3

333202_0905.qxd 12/5/05 11:37 AM Page 685

Binomial ExpansionsAs mentioned at the beginning of this section, when you write out the coeffi-cients for a binomial that is raised to a power, you are expanding a binomial.The formulas for binomial coefficients give you an easy way to expand binomi-als, as demonstrated in the next four examples.

Expanding a Binomial

Write the expansion for the expression

SolutionThe binomial coefficients from the third row of Pascal’s Triangle are

So, the expansion is as follows.


To expand binomials representing differences rather than sums, you alternatesigns. Here are two examples.


Write the expansion for each expression.

a. b.

SolutionThe binomial coefficients from the fourth row of Pascal’s Triangle are

Therefore, the expansions are as follows.

a.

b.


� x4 � 8x3y � 24x2y2 � 32xy3 � 16y4

�x � 2y�4 � �1�x4 � �4�x3�2y� � �6�x2�2y�2 � �4�x�2y�3 � �1��2y�4

� 16x4 � 96x3 � 216x2 � 216x � 81

�2x � 3�4 � �1��2x�4 � �4��2x�3�3� � �6��2x�2�32� � �4��2x��33� � �1��34�

1, 4, 6, 4, 1.

�x � 2y�4�2x � 3�4

�x � 1�4 � x4 � 4x3 � 6x 2 � 4x � 1

�x � 1�3 � x3 � 3x2 � 3x � 1

� x3 � 3x2 � 3x � 1

�x � 1�3 � �1�x3 � �3�x2�1� � �3�x�12� � �1��13�

1, 3, 3, 1.

�x � 1�3.


Historical NotePrecious Mirror “Pascal’s”Triangleand forms of the BinomialTheorem were known inEastern cultures prior to the Western “discovery” of the theorem. A Chinese text entitled Precious Mirrorcontains a triangle of binomialexpansions through the eighthpower.

Example 4

Example 5

333202_0905.qxd 12/5/05 11:37 AM Page 686


Write the expansion for

SolutionUse the third row of Pascal’s Triangle, as follows.


Sometimes you will need to find a specific term in a binomial expansion.Instead of writing out the entire expansion, you can use the fact that, from theBinomial Theorem, the term is

Finding a Term in a Binomial Expansion

a. Find the sixth term of

b. Find the coefficient of the term in the expansion of

Solutiona. Remember that the formula is for the term, so is one less than the

number of the term you are looking for. So, to find the sixth term in thisbinomial expansion, use and as shown.

b. In this case, and Substitute these values toobtain

So, the coefficient is


�10,777,536.

� �10,777,536a6b5.

� �462��729a6��32b5�

nCr xn�r yr � 11C5�3a�6��2b�5

y � �2b.x � 3a,r � 5,n � 11,

� 1792a3b5.� 56�25�a3b5 8C5a8�5�2b�5 � 56 � a3 � �2b�5

y � 2b,x � a,n � 8,r � 5,

r�r � 1�th

�3a � 2b�11.a6b5

�a � 2b�8.

nCr xn�r yr.�r � 1�th

� x6 � 12x4 � 48x2 � 64

�x2 � 4�3 � �1��x2�3 � �3��x2�2�4� � �3�x2�42� � �1��43�

�x2 � 4�3.


Example 6

Example 7


Error Analysis You are a math instructor and receive the following solutions fromone of your students on a quiz. Find the error(s) in each solution. Discuss ways thatyour student could avoid the error(s) in the future.

a. Find the second term in the expansion of

b. Find the fourth term in the expansion of

6C4�12 x�2�7y�4 � 9003.75x 2y 4

�12 x � 7y�6

.

5�2x�4�3y�2 � 720x 4y 2

�2x � 3y�5.

You can use a graphing utility tocheck the expansion in Example 6.Graph the original binomialexpression and the expansion inthe same viewing window. Thegraphs should coincide as shownbelow.

−100

−5 5

200

Techno logy

333202_0905.qxd 12/5/05 11:37 AM Page 687


Exercises 9.5

In Exercises 1–10, calculate the binomial coefficient.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

In Exercises 11–14, evaluate using Pascal’s Triangle.

11. 12.

13. 14.

In Exercises 15–34, use the Binomial Theorem to expandand simplify the expression.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33.

34.

In Exercises 35–38, expand the binomial by using Pascal’sTriangle to determine the coefficients.

35. 36.

37. 38.

In Exercises 39– 46, find the specified th term in theexpansion of the binomial.

39. 40.

41. 42.

43. 44.

45. 46.

In Exercises 47–54, find the coefficient a of the term in theexpansion of the binomial.

Binomial Term

47.

48.

49.

50.

51.

52.

53.

54.

In Exercises 55–58, use the Binomial Theorem to expandand simplify the expression.

55.

56.

57.

58.

In Exercises 59–62, expand the expression in the differencequotient and simplify.

Difference quotient

59. 60.

61. 62. f �x� �1

xf �x� � �x

f �x� � x4f �x� � x3

f �x � h� � f �x�h

�u3�5 � 2�5

�x 2�3 � y1�3�3

�2�t � 1�3��x � 3�4

az4t8�z 2 � t�10

ax8y 6�x 2 � y�10

ax 6y 2�2x � 3y�8

ax4y5�3x � 2y�9

ax 2y8�4x � y�10

ax8y 2�x � 2y�10

ax8�x 2 � 3�12

ax5�x � 3�12

n � 7�7x � 2y�15,n � 9�10x � 3y�12,

n � 5�5a � 6b�5,n � 8�4x � 3y�9,

n � 4�x � 10z�7,n � 3�x � 6y�5,

n � 7�x � y�6,n � 4�x � y�10,

n

�2v � 3�6�x � 2y�5

�3 � 2z�4�2t � s�5

3�x � 1�5 � 4�x � 1�3

2�x � 3�4 � 5�x � 3�2

�1

x� 2y�

6

�1

x� y�

5

�x 2 � y 2�6�x 2 � y2�4

�7a � b�3�2x � y�3

�2x � 5y�5�3a � 4b�5

�x � 2y�4�r � 3s�6

�c � d�3�x � y�5

�y � 2�5�y � 4�3

�a � 5�5�a � 6�4

�x � 1�6�x � 1�4

6C37C4

�87��8

5�

�1002 ��100

98 ��10

6 ��104 �

12C520C15

20C2012C0

8C65C3


1. The coefficients of a binomial expansion are called ________ ________.

2. To find binomial coefficients, you can use the ________ ________ or ________ ________.

3. The notation used to denote a binomial coefficient is ________ or ________.

4. When you write out the coefficients for a binomial that is raised to a power, you are ________ a ________.


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In Exercises 63–68, use the Binomial Theorem to expandthe complex number. Simplify your result.

63. 64.

65. 66.

67. 68.

Approximation In Exercises 69–72, use the BinomialTheorem to approximate the quantity accurate to threedecimal places. For example, in Exercise 69, use theexpansion

69. 70.

71. 72.

Graphical Reasoning In Exercises 73 and 74, use a graph-ing utility to graph and in the same viewing window.What is the relationship between the two graphs? Use theBinomial Theorem to write the polynomial function instandard form.

73.

74.

Probability In Exercises 75–78, consider independenttrials of an experiment in which each trial has two possibleoutcomes: “success” or “failure.” The probability of asuccess on each trial is and the probability of a failure is

In this context, the term in theexpansion of gives the probability of successesin the trials of the experiment.

75. A fair coin is tossed seven times. To find the probability ofobtaining four heads, evaluate the term

in the expansion of

76. The probability of a baseball player getting a hit during anygiven time at bat is To find the probability that the playergets three hits during the next 10 times at bat, evaluate theterm

in the expansion of

77. The probability of a sales representative making a sale withany one customer is The sales representative makes eightcontacts a day. To find the probability of making four sales,evaluate the term

in the expansion of

78. To find the probability that the sales representative inExercise 77 makes four sales if the probability of a salewith any one customer is evaluate the term

in the expansion of � 12 �12�8

.8C4�1

2�4�12�4

12,

�13 �

23�8

.

8C4�13�4�2

3�4

13.

�14 �

34�10

.

10C3�14�3�3

4�7

14.

� 12 �12�7

.

7C4�12�4� 12�3

nk� p � q�n

nC k p k q n� kq � 1 � p.p,

n

g�x� � f �x � 3�f �x� � �x4 � 4x 2 � 1,

g�x� � f �x � 4�f �x� � x3 � 4x,

g

gf

�1.98�9�2.99�12

�2.005�10�1.02�8

�1.02�8 � �1 � 0.02�8 �1� 8�0.02�� 28�0.02�2 � . . . .

�5 � �3 i�4��1

2�

�3

2i�

3

�5 � ��9 �3�2 � 3i�6

�2 � i�5�1 � i�4

79. Data Analysis: Water Consumption The tableshows the per capita consumption of bottled water (in gallons) in the United States from 1990 through2003. (Source: Economic Research Service, U.S.Department of Agriculture)

(a) Use the regression feature of a graphing utility tofind a cubic model for the data. Let represent theyear, with corresponding to 1990.

(b) Use a graphing utility to plot the data and themodel in the same viewing window.

(c) You want to adjust the model so that corre-sponds to 2000 rather than 1990. To do this, youshift the graph of 10 units to the left to obtain

Write in standard form.

(d) Use a graphing utility to graph in the sameviewing window as

(e) Use both models to estimate the per capitaconsumption of bottled water in 2008. Do youobtain the same answer?

(f) Describe the overall trend in the data. What factorsdo you think may have contributed to the increasein the per capita consumption of bottled water?

f.g

g�t�g�t� � f �t � 10�.f

t � 0

t � 0t

f �t�

Model It

Year Consumption,

1990 8.0

1991 8.0

1992 9.7

1993 10.3

1994 11.3

1995 12.1

1996 13.0

1997 13.9

1998 15.0

1999 16.4

2000 17.4

2001 18.8

2002 20.7

2003 22.0

f �t�

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80. Child Support The amounts (in billions of dollars)of child support collected in the United States from 1990 to2002 can be approximated by the model

where represents the year, with corresponding to1990 (see figure). (Source: U.S. Department of Healthand Human Services)

(a) You want to adjust the model so that correspondsto 2000 rather than 1990. To do this, you shift the graphof 10 units to the left and obtain Write in standard form.

(b) Use a graphing utility to graph and in the sameviewing window.

(c) Use the graphs to estimate when the child support col-lections will exceed $30 billion.

Synthesis

True or False? In Exercises 81– 83, determine whether thestatement is true or false. Justify your answer.

81. The Binomial Theorem could be used to produce each rowof Pascal’s Triangle.

82. A binomial that represents a difference cannot always beaccurately expanded using the Binomial Theorem.

83. The -term and the -term of the expansion ofhave identical coefficients.

84. Writing In your own words, explain how to form therows of Pascal’s Triangle.

85. Form rows 8–10 of Pascal’s Triangle.

86. Think About It How many terms are in the expansion of

87. Think About It How do the expansions of anddiffer?

88. Graphical Reasoning Which two functions haveidentical graphs, and why? Use a graphing utility to graphthe functions in the given order and in the same viewingwindow. Compare the graphs.

(a)

(b)

(c)

(d)

(e)

Proof In Exercises 89–92, prove the property for allintegers and where

89.

90.

91.

92. The sum of the numbers in the th row of Pascal’s Triangleis

Skills Review

In Exercises 93–96, the graph of is shown. Graph and use the graph to write an equation for the graph of

93. 94.

95. 96.

In Exercises 97 and 98, find the inverse of the matrix.

97. 98. 1.2�2

�2.34�6

�554

x1 4−1

1

2 3

−4

−5

y

x1 3

5

3

4

−1−2

1

2

y

f �x� � �xf �x� � �x

x1 2

−2

234

−1

−3

−2−3

y

x1 32

1

−1

23456

4 5 6

y

f �x� � x2f �x� � x2

g.fy � g�x�

2n.n

n�1Cr � nCr �nCr �1

nC0 �nC1 �nC2 � . . . ± nCn � 0nCr � nCn �r

0 ≤ r ≤ n.nr

p�x� � 1 � 3x � 3x 2 � x 3

k�x� � 1 � 3x � 3x 2 � x 3

h�x� � 1 � 3x � 3x 2 � x3

g�x� � 1 � x3

f �x� � �1 � x�3

�x � y�n

�x � y�n

�x � y�n?

�x2 � 3�12x14x10

gf

g�t�g�t� � f �t � 10�.f

t � 0

Chi

ld s

uppo

rt c

olle

ctio

ns(i

n bi

llion

s of

dol

lars

)

t

f(t)

Year (0 ↔ 1990)1 2 3 4 5 6 7 8 9 10 11 12 13

369

121518212427

t � 0t

0 ≤ t ≤ 12f �t� � 0.031t 2 � 0.82t � 6.1,

f �t�


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Section 9.6 Counting Principles 691

Simple Counting ProblemsThis section and Section 9.7 present a brief introduction to some of the basiccounting principles and their application to probability. In Section 9.7, you willsee that much of probability has to do with counting the number of ways an eventcan occur. The following two examples describe simple counting problems.

Selecting Pairs of Numbers at Random

Eight pieces of paper are numbered from 1 to 8 and placed in a box. One pieceof paper is drawn from the box, its number is written down, and the piece ofpaper is replaced in the box. Then, a second piece of paper is drawn from thebox, and its number is written down. Finally, the two numbers are added togeth-er. How many different ways can a sum of 12 be obtained?

SolutionTo solve this problem, count the different ways that a sum of 12 can be obtainedusing two numbers from 1 to 8.

First number 4 5 6 7 8

Second number 8 7 6 5 4

From this list, you can see that a sum of 12 can occur in five different ways.

Now try Exercise 5.

Selecting Pairs of Numbers at Random

Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two piecesof paper are drawn from the box at the same time, and the numbers on the piecesof paper are written down and totaled. How many different ways can a sum of 12be obtained?

SolutionTo solve this problem, count the different ways that a sum of 12 can be obtainedusing two different numbers from 1 to 8.

First number 4 5 7 8

Second number 8 7 5 4

So, a sum of 12 can be obtained in four different ways.

Now try Exercise 7.

The difference between the counting problems in Examples 1 and 2 can bedescribed by saying that the random selection in Example 1 occurs withreplacement, whereas the random selection in Example 2 occurs withoutreplacement, which eliminates the possibility of choosing two 6’s.

What you should learn• Solve simple counting

problems.

• Use the FundamentalCounting Principle to solvecounting problems.

• Use permutations to solvecounting problems.

• Use combinations to solvecounting problems.

Why you should learn itYou can use counting principlesto solve counting problems thatoccur in real life. For instance, inExercise 65 on page 700, you areasked to use counting principlesto determine the number ofpossible ways of selecting thewinning numbers in thePowerball lottery.

Counting Principles

© Michael Simpson/FPG/Getty Images

9.6

Example 1

Example 2

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The Fundamental Counting PrincipleExamples 1 and 2 describe simple counting problems in which you can list eachpossible way that an event can occur. When it is possible, this is always the bestway to solve a counting problem. However, some events can occur in so manydifferent ways that it is not feasible to write out the entire list. In such cases, youmust rely on formulas and counting principles. The most important of these is theFundamental Counting Principle.

The Fundamental Counting Principle can be extended to three or moreevents. For instance, the number of ways that three events and canoccur is

Using the Fundamental Counting Principle

How many different pairs of letters from the English alphabet are possible?

SolutionThere are two events in this situation. The first event is the choice of the firstletter, and the second event is the choice of the second letter. Because the Englishalphabet contains 26 letters, it follows that the number of two-letter pairs is


Using the Fundamental Counting Principle

Telephone numbers in the United States currently have 10 digits. The first threeare the area code and the next seven are the local telephone number. How manydifferent telephone numbers are possible within each area code? (Note that at thistime, a local telephone number cannot begin with 0 or 1.)

SolutionBecause the first digit of a local telephone number cannot be 0 or 1, there areonly eight choices for the first digit. For each of the other six digits, there are 10choices.

So, the number of local telephone numbers that are possible within each areacode is


8 � 10 � 10 � 10 � 10 � 10 � 10 � 8,000,000.

Local Number

101010 1010108

Area Code

26 � 26 � 676.

m1 � m2 � m3.E3E1, E2,


Fundamental Counting PrincipleLet and be two events. The first event can occur in differentways. After has occurred, can occur in different ways. The numberof ways that the two events can occur is m1 � m2.

m2E2E1

m1E1E2E1

Example 3

Example 4

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PermutationsOne important application of the Fundamental Counting Principle is indetermining the number of ways that elements can be arranged (in order). Anordering of elements is called a permutation of the elements.

Finding the Number of Permutations of n Elements

How many permutations are possible for the letters A, B, C, D, E, and F?

SolutionConsider the following reasoning.

First position: Any of the six letters

Second position: Any of the remaining five letters

Third position: Any of the remaining four letters

Fourth position: Any of the remaining three letters

Fifth position: Any of the remaining two letters

Sixth position: The one remaining letter

So, the numbers of choices for the six positions are as follows.

The total number of permutations of the six letters is


� 720.

6! � 6 � 5 � 4 � 3 � 2 � 1

Permutations of six letters

6 5 4 3 2 1

nn


Definition of PermutationA permutation of different elements is an ordering of the elements suchthat one element is first, one is second, one is third, and so on.

n

Number of Permutations of n ElementsThe number of permutations of elements is

In other words, there are different ways that elements can be ordered.nn!

n � �n � 1� . . . 4 � 3 � 2 � 1 � n!.

n

Example 5

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Counting Horse Race Finishes

Eight horses are running in a race. In how many different ways can these horsescome in first, second, and third? (Assume that there are no ties.)

SolutionHere are the different possibilities.

Win (first position): Eight choices

Place (second position): Seven choices

Show (third position): Six choices

Using the Fundamental Counting Principle, multiply these three numberstogether to obtain the following.

So, there are different orders.


It is useful, on occasion, to order a subset of a collection of elements ratherthan the entire collection. For example, you might want to choose and order

elements out of a collection of elements. Such an ordering is called apermutation of n elements taken r at a time.

Using this formula, you can rework Example 6 to find that the number ofpermutations of eight horses taken three at a time is

which is the same answer obtained in the example.

� 336

�8 � 7 � 6 � 5!

5!

�8!5!

8P3 �8!

�8 � 3�!

nr

8 � 7 � 6 � 336

Different orders of horses

8 7 6


Permutations of n Elements Taken r at a TimeThe number of permutations of elements taken at a time is

� n�n � 1��n � 2� . . . �n � r � 1�.

nPr �n!

�n � r�!

rn

Example 6

Most graphing calculators are programmed to evaluate Consult the user’s guide for yourcalculator and then evaluate You should get an answer of 6720.

8 P5.

nPr.

Techno logy

Eleven thoroughbred racehorseshold the title of Triple Crownwinner for winning the KentuckyDerby, the Preakness, and theBelmont Stakes in the same year.Forty-nine horses have won twoout of the three races.

Vau

gh

n Y

ou

tz/N

ewsm

aker

s/G

etty

Imag

es

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Remember that for permutations, order is important. So, if you are lookingat the possible permutations of the letters A, B, C, and D taken three at a time,the permutations (A, B, D) and (B, A, D) are counted as different because theorder of the elements is different.

Suppose, however, that you are asked to find the possible permutations ofthe letters A, A, B, and C. The total number of permutations of the fourletters would be However, not all of these arrangements would bedistinguishable because there are two A’s in the list. To find the number ofdistinguishable permutations, you can use the following formula.

Distinguishable Permutations

In how many distinguishable ways can the letters in BANANA be written?

SolutionThis word has six letters, of which three are A’s, two are N’s, and one is a B. So,the number of distinguishable ways the letters can be written is

The 60 different distinguishable permutations are as follows.


AAABNNAANABNABAANNANAABNANBAANBAAANNBNAAANNAABANNABNAANBANAA

AAANBNAANANBABANANANAANBANBANABAANANBNAANANAABNANANAABNBNAAA

AAANNBAANBANABANNAANABANANBNAABAANNABNANAANAANABNANABANNAAAB

AABANNAANBNAABNAANANABNAANNAABBANAANBNNAAANAANBANANBAANNAABA

AABNANAANNABABNANAANANABANNABABANANANAAABNNABAANNBAAANNNABAA

AABNNAAANNBAABNNAAANANBAANNBAABANNAANAAANBNABANANBAANANNBAAA

� 60.

�6 � 5 � 4 � 3!

3! � 2!

n!

n1! � n2! � n3!�

6!

3! � 2! � 1!

4P4 � 4!.


Distinguishable PermutationsSuppose a set of objects has of one kind of object, of a second kind,

of a third kind, and so on, with

Then the number of distinguishable permutations of the objects is

n!

n1! � n2! � n3! � . . . � nk !.

n

n3 � . . . � nk .n � n1 � n2 �

n3

n2n1n

Example 7

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CombinationsWhen you count the number of possible permutations of a set of elements, orderis important. As a final topic in this section, you will look at a method of selectingsubsets of a larger set in which order is not important. Such subsets are calledcombinations of n elements taken r at a time. For instance, the combinations

and

are equivalent because both sets contain the same three elements, and the orderin which the elements are listed is not important. So, you would count only oneof the two sets. A common example of how a combination occurs is a card gamein which the player is free to reorder the cards after they have been dealt.

Combinations of n Elements Taken r at a Time

In how many different ways can three letters be chosen from the letters A, B, C,D, and E? (The order of the three letters is not important.)

SolutionThe following subsets represent the different combinations of three letters thatcan be chosen from the five letters.

From this list, you can conclude that there are 10 different ways that three letterscan be chosen from five letters.


Note that the formula for is the same one given for binomial coefficients.To see how this formula is used, solve the counting problem in Example 8. In thatproblem, you are asked to find the number of combinations of five elementstaken three at a time. So, and the number of combinations is

which is the same answer obtained in Example 8.

5C3 �5!

2!3!�

5 � 4 � 3!

2 � 1 � 3!� 10

2

n � 5, r � 3,

nCr

�C, D, E��B, D, E��B, C, E��B, C, D��A, D, E��A, C, E��A, C, D��A, B, E��A, B, D��A, B, C�

�B, A, C��A, B, C�


Combinations of n Elements Taken r at a TimeThe number of combinations of elements taken at a time is

which is equivalent to nCr � nPr

r!.nCr �

n!�n � r�!r!

rn

Example 8

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Counting Card Hands

A standard poker hand consists of five cards dealt from a deck of 52 (see Figure9.7). How many different poker hands are possible? (After the cards are dealt, theplayer may reorder them, and so order is not important.)

SolutionYou can find the number of different poker hands by using the formula for thenumber of combinations of 52 elements taken five at a time, as follows.


Forming a Team

You are forming a 12-member swim team from 10 girls and 15 boys. The teammust consist of five girls and seven boys. How many different 12-member teamsare possible?

SolutionThere are ways of choosing five girls. The are ways of choosing sevenboys. By the Fundamental Counting Principal, there are ways ofchoosing five girls and seven boys.

So, there are 1,621,620 12-member swim teams possible.


When solving problems involving counting principles, you need to be ableto distinguish among the various counting principles in order to determine whichis necessary to solve the problem correctly. To do this, ask yourself the follow-ing questions.

1. Is the order of the elements important? Permutation

2. Are the chosen elements a subset of a larger set in which order is notimportant? Combination

3. Does the problem involve two or more separate events? FundamentalCounting Principle

� 1,621,620

� 252 � 6435

10C5 � 15C7 �10!

5! � 5!�

15!8! � 7!

10C5 � 15C7

15C710C5

� 2,598,960

�52 � 51 � 50 � 49 � 48 � 47!

5 � 4 � 3 � 2 � 1 � 47!

�52!

47!5!

52C5 �52!

�52 � 5�!5!


A

2

3

4

5

6

7

8

9

10

J

Q

K

A

2

3

4

5

6

7

8

9

10

J

Q

K

A

2

3

4

5

6

7

8

9

10

J

Q

K

A

2

3

4

5

6

7

8

9

10

J

Q

K

Example 9

Example 10

FIGURE 9.7 Standard deck of playingcards

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Random Selection In Exercises 1– 8, determine thenumber of ways a computer can randomly generate one ormore such integers from 1 through 12.

1. An odd integer 2. An even integer

3. A prime integer

4. An integer that is greater than 9

5. An integer that is divisible by 4

6. An integer that is divisible by 3

7. Two distinct integers whose sum is 9

8. Two distinct integers whose sum is 8

9. Entertainment Systems A customer can choose one ofthree amplifiers, one of two compact disc players, and oneof five speaker models for an entertainment system.Determine the number of possible system configurations.

10. Job Applicants A college needs two additional facultymembers: a chemist and a statistician. In how many wayscan these positions be filled if there are five applicants forthe chemistry position and three applicants for the statisticsposition?

11. Course Schedule A college student is preparing a courseschedule for the next semester. The student may select oneof two mathematics courses, one of three science courses,and one of five courses from the social sciences andhumanities. How many schedules are possible?

12. Aircraft Boarding Eight people are boarding an aircraft.Two have tickets for first class and board before those inthe economy class. In how many ways can the eight peopleboard the aircraft?

13. True-False Exam In how many ways can a six-questiontrue-false exam be answered? (Assume that no questionsare omitted.)

14. True-False Exam In how many ways can a 12-questiontrue-false exam be answered? (Assume that no questionsare omitted.)

15. License Plate Numbers In the state of Pennsylvania,each standard automobile license plate number consists ofthree letters followed by a four-digit number. How manydistinct license plate numbers can be formed inPennsylvania?

16. License Plate Numbers In a certain state, each automo-bile license plate number consists of two letters followedby a four-digit number. To avoid confusion between “O”and “zero” and between “I” and “one,” the letters “O” and“I” are not used. How many distinct license plate numberscan be formed in this state?

17. Three-Digit Numbers How many three-digit numberscan be formed under each condition?

(a) The leading digit cannot be zero.

(b) The leading digit cannot be zero and no repetition ofdigits is allowed.

(c) The leading digit cannot be zero and the number mustbe a multiple of 5.

(d) The number is at least 400.

18. Four-Digit Numbers How many four-digit numbers canbe formed under each condition?

(a) The leading digit cannot be zero.

(b) The leading digit cannot be zero and no repetition ofdigits is allowed.

(c) The leading digit cannot be zero and the number mustbe less than 5000.

(d) The leading digit cannot be zero and the number mustbe even.

19. Combination Lock A combination lock will open whenthe right choice of three numbers (from 1 to 40, inclusive)is selected. How many different lock combinations arepossible?


Exercises 9.6


1. The ________ ________ ________ states that if there are ways for one event to occur and ways for a second event to occur, there are ways for both events to occur.

2. An ordering of elements is called a ________ of the elements.

3. The number of permutations of elements taken at a time is given by the formula ________.

4. The number of ________ ________ of objects is given by

5. When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of elements taken at a time.


rn

n!n1!n2!n3! . . . nk!

.n

rn

n

m1 � m2

m2m1

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www.Eduspace.com


20. Combination Lock A combination lock will open whenthe right choice of three numbers (from 1 to 50, inclusive)is selected. How many different lock combinations arepossible?

21. Concert Seats Four couples have reserved seats in a rowfor a concert. In how many different ways can they beseated if

(a) there are no seating restrictions?

(b) the two members of each couple wish to sit together?

22. Single File In how many orders can four girls and fourboys walk through a doorway single file if

(a) there are no restrictions?

(b) the girls walk through before the boys?

In Exercises 23–28, evaluate

23. 24.

25. 26.

27. 28.

In Exercises 29 and 30, solve for

29. 30.

In Exercises 31–36, evaluate using a graphing utility.

31. 32.

33. 34.

35. 36.

37. Posing for a Photograph In how many ways can fivechildren posing for a photograph line up in a row?

38. Riding in a Car In how many ways can six people sit ina six-passenger car?

39. Choosing Officers From a pool of 12 candidates, theoffices of president, vice-president, secretary, and treasurerwill be filled. In how many different ways can the officesbe filled?

40. Assembly Line Production There are four processesinvolved in assembling a product, and these processes canbe performed in any order. The management wants to testeach order to determine which is the least time-consuming.How many different orders will have to be tested?

In Exercises 41–44, find the number of distinguishablepermutations of the group of letters.

41. A, A, G, E, E, E, M 42. B, B, B, T, T, T, T, T

43. A, L, G, E, B, R, A 44. M, I, S, S, I, S, S, I, P, P, I

45. Write all permutations of the letters A, B, C, and D.

46. Write all permutations of the letters A, B, C, and D if theletters B and C must remain between the letters A and D.

47. Batting Order A baseball coach is creating a nine-playerbatting order by selecting from a team of 15 players. Howmany different batting orders are possible?

48. Athletics Six sprinters have qualified for the finals in the100-meter dash at the NCAA national track meet. In howmany ways can the sprinters come in first, second, andthird? (Assume there are no ties.)

49. Jury Selection From a group of 40 people, a jury of 12people is to be selected. In how many different ways canthe jury be selected?

50. Committee Members As of January 2005, the U.S.Senate Committee on Indian Affairs had 14 members.Assuming party affiliation was not a factor in selection,how many different committees were possible from the100 U.S. senators?

51. Write all possible selections of two letters that can beformed from the letters A, B, C, D, E, and F. (The order ofthe two letters is not important.)

52. Forming an Experimental Group In order to conduct anexperiment, five students are randomly selected from aclass of 20. How many different groups of five students arepossible?

53. Lottery Choices In the Massachusetts Mass Cash game,a player chooses five distinct numbers from 1 to 35. In howmany ways can a player select the five numbers?

54. Lottery Choices In the Louisiana Lotto game, a playerchooses six distinct numbers from 1 to 40. In how manyways can a player select the six numbers?

55. Defective Units A shipment of 10 microwave ovenscontains three defective units. In how many ways can avending company purchase four of these units and receive(a) all good units, (b) two good units, and (c) at least twogood units?

56. Interpersonal Relationships The complexity of interper-sonal relationships increases dramatically as the size of agroup increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3,(b) 8, (c) 12, and (d) 20.

57. Poker Hand You are dealt five cards from an ordinarydeck of 52 playing cards. In how many ways can you get(a) a full house and (b) a five-card combination containingtwo jacks and three aces? (A full house consists of three ofone kind and two of another. For example, A-A-A-5-5 andK-K-K-10-10 are full houses.)

58. Job Applicants A toy manufacturer interviews eightpeople for four openings in the research and developmentdepartment of the company. Three of the eight people arewomen. If all eight are qualified, in how many ways canthe employer fill the four positions if (a) the selection israndom and (b) exactly two selections are women?

10C720C5

10 P8100 P3

100 P520 P5

nP5 � 18 � n�2P414 � nP3 � n�2P4

n.

7P45P4

20 P28P3

5 P54P4

n Pr .

333202_0906.qxd 12/5/05 11:39 AM Page 699

59. Forming a Committee A six-member researchcommittee at a local college is to be formed having oneadministrator, three faculty members, and two students.There are seven administrators, 12 faculty members, and20 students in contention for the committee. How manysix-member committees are possible?

60. Law Enforcement A police department uses computerimaging to create digital photographs of alleged perpetra-tors from eyewitness accounts. One software packagecontains 195 hairlines, 99 sets of eyes and eyebrows,89 noses, 105 mouths, and 74 chins and cheek structures.

(a) Find the possible number of different faces that thesoftware could create.

(b) A eyewitness can clearly recall the hairline and eyesand eyebrows of a suspect. How many different facescan be produced with this information?

Geometry In Exercises 61–64, find the number of diago-nals of the polygon. (A line segment connecting any twononadjacent vertices is called a diagonal of the polygon.)

61. Pentagon 62. Hexagon

63. Octagon 64. Decagon (10 sides)

66. Permutations or Combinations? Decide whether eachscenario should be counted using permutations orcombinations. Explain your reasoning.

(a) Number of ways 10 people can line up in a row for con-cert tickets

(b) Number of different arrangements of three types offlowers from an array of 20 types

(c) Number of three-digit pin numbers for a debit card

(d) Number of two-scoop ice cream cones created from 31different flavors

Synthesis


67. The number of letter pairs that can be formed in any orderfrom any of the first 13 letters in the alphabet (A–M) is anexample of a permutation.

68. The number of permutations of elements can be deter-mined by using the Fundamental Counting Principle.

69. What is the relationship between and ?

70. Without calculating the numbers, determine which of thefollowing is greater. Explain.

(a) The number of combinations of 10 elements taken sixat a time

(b) The number of permutations of 10 elements taken sixat a time

Proof In Exercises 71– 74, prove the identity.

71. 72.

73. 74.

75. Think About It Can your calculator evaluate Ifnot, explain why.

76. Writing Explain in words the meaning of

Skills Review

In Exercises 77– 80, evaluate the function at each specifiedvalue of the independent variable and simplify.

77.

(a) (b) (c)

78.

(a) (b) (c)

79.

(a) (b) (c)

80.

(a) (b) (c)

In Exercises 81– 84, solve the equation. Round your answerto two decimal places, if necessary.

81. 82.

83. 84. e x�3 � 16log2�x � 3� � 5

4t

�32t

� 1�x � 3 � x � 6

f ��20�f ��1�f ��4�

�x2 � 2x � 5,

�x2 � 2,

x ≤ �4

x > �4f �x� �

f �11�f ��1�f ��5�f �x� � ��x � 5� � 6

g�x � 1�g�7�g�3�g�x� � �x � 3 � 2

f ��5�f �0�f �3�f �x� � 3x2 � 8

n Pr .

100P80?

nCr � n Pr

r!nCn �1 � nC1

nCn �nC0n Pn �1 � n Pn

nCn�rnCr

n


65. Lottery Powerball is a lottery game that is operated bythe Multi-State Lottery Association and is played in 27states, Washington D.C., and the U.S. Virgin Islands.The game is played by drawing five white balls out of adrum of 53 white balls (numbered 1–53) and one redpowerball out of a drum of 42 red balls (numbered1– 42). The jackpot is won by matching all five whiteballs in any order and the red powerball.

(a) Find the possible number of winning Powerballnumbers.

(b) Find the possible number of winning Powerballnumbers if the jackpot is won by matching all fivewhite balls in order and the red power ball.

(c) Compare the results of part (a) with a state lotteryin which a jackpot is won by matching six ballsfrom a drum of 53 balls.

Model It

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Section 9.7 Probability 701

The Probability of an EventAny happening for which the result is uncertain is called an experiment. Thepossible results of the experiment are outcomes, the set of all possible outcomesof the experiment is the sample space of the experiment, and any subcollectionof a sample space is an event.

For instance, when a six-sided die is tossed, the sample space can be repre-sented by the numbers 1 through 6. For this experiment, each of the outcomes isequally likely.

To describe sample spaces in such a way that each outcome is equally likely,you must sometimes distinguish between or among various outcomes in waysthat appear artificial. Example 1 illustrates such a situation.

Finding a Sample Space

Find the sample space for each of the following.

a. One coin is tossed.

b. Two coins are tossed.

c. Three coins are tossed.

Solutiona. Because the coin will land either heads up (denoted by ) or tails up (denoted

by ), the sample space is

b. Because either coin can land heads up or tails up, the possible outcomes areas follows.

heads up on both coins

heads up on first coin and tails up on second coin

tails up on first coin and heads up on second coin

tails up on both coins

So, the sample space is

Note that this list distinguishes between the two cases and eventhough these two outcomes appear to be similar.

c. Following the notation of part (b), the sample space is

Note that this list distinguishes among the cases HHT, HTH, and THH, andamong the cases HTT, THT, and TTH.

Now try Exercise 1.

S � �HHH, HHT, HTH, HTT, THH, THT, TTH, TTT �.

TH,HT

S � �HH, HT, TH, TT �.

T T �

TH �

HT �

HH �

S � �H, T �.

TH

What you should learn• Find the probabilities of

events.

• Find the probabilities of mutually exclusive events.

• Find the probabilities of independent events.

• Find the probability of thecomplement of an event.

Why you should learn itProbability applies to manygames of chance. For instance,in Exercise 55, on page 712, youwill calculate probabilities thatrelate to the game of roulette.

Probability

Hank de Lespinasse/The Image Bank

9.7

Example 1

333202_0907.qxd 12/5/05 11:41 AM Page 701

To calculate the probability of an event, count the number of outcomes in theevent and in the sample space. The number of outcomes in event is denoted by

and the number of outcomes in the sample space is denoted by Theprobability that event will occur is given by

Because the number of outcomes in an event must be less than or equal tothe number of outcomes in the sample space, the probability of an event must bea number between 0 and 1. That is,

as indicated in Figure 9.8. If event cannot occur, and is called animpossible event. If event must occur, and is called a certainevent.

Finding the Probability of an Event

a. Two coins are tossed. What is the probability that both land heads up?

b. A card is drawn from a standard deck of playing cards. What is the probabilitythat it is an ace?

Solutiona. Following the procedure in Example 1(b), let

and

The probability of getting two heads is

b. Because there are 52 cards in a standard deck of playing cards and there arefour aces (one in each suit), the probability of drawing an ace is


�1

13.

�4

52

P�E� �n�E�n�S�

�1

4.P�E� �

n�E�n�S�

S � �HH, HT, TH, TT �.

E � �HH�

EEP�E� � 1,EEP�E� � 0,

0 ≤ P�E� ≤ 1

n�E��n�S�.En�S�.Sn�E�,

E


Toss two coins 100 times andwrite down the number ofheads that occur on each toss(0, 1, or 2). How many timesdid two heads occur? Howmany times would you expecttwo heads to occur if you didthe experiment 1000 times?

Exploration

You can write a probability as a fraction, decimal, or percent.For instance, in Example 2(a),the probability of getting twoheads can be written as 0.25,or 25%.

14,

Impossibleevent

(cannotoccur)

Increasing likelihoodof occurrence

0.0 0.5 1.0

The occurrenceof the event is

just as likely asit is unlikely.

Certainevent(mustoccur)

The Probability of an EventIf an event has equally likely outcomes and its sample space has

equally likely outcomes, the probability of event is


.

En�S�Sn�E�E

Example 2 FIGURE 9.8

333202_0907.qxd 12/5/05 11:41 AM Page 702


FIGURE 9.9

FIGURE 9.10


Two six-sided dice are tossed. What is the probability that the total of the two diceis 7? (See Figure 9.9.)

SolutionBecause there are six possible outcomes on each die, you can use theFundamental Counting Principle to conclude that there are or 36 differentoutcomes when two dice are tossed. To find the probability of rolling a total of 7,you must first count the number of ways in which this can occur.

So, a total of 7 can be rolled in six ways, which means that the probability ofrolling a 7 is



Twelve-sided dice, as shown in Figure 9.10, can be constructed (in the shape ofregular dodecahedrons) such that each of the numbers from 1 to 6 appears twiceon each die. Prove that these dice can be used in any game requiring ordinary six-sided dice without changing the probabilities of different outcomes.

SolutionFor an ordinary six-sided die, each of the numbers 1, 2, 3, 4, 5, and 6 occurs onlyonce, so the probability of any particular number coming up is

For one of the 12-sided dice, each number occurs twice, so the probability of anyparticular number coming up is



�2

12�

1

6.


�1

6.

�1

6.�

6

36P�E� �

n�E�n�S�

6 � 6

Example 3

First die Second die

1 6

2 5

3 4

4 3

5 2

6 1

Example 4

You could have written out eachsample space in Examples 2 and3 and simply counted the out-comes in the desired events. Forlarger sample spaces, however,you should use the counting prin-ciples discussed in Section 9.6.

333202_0907.qxd 12/5/05 11:41 AM Page 703

The Probability of Winning a Lottery

In the Arizona state lottery, a player chooses six different numbers from 1 to 41.If these six numbers match the six numbers drawn (in any order) by the lotterycommission, the player wins (or shares) the top prize. What is the probability ofwinning the top prize if the player buys one ticket?

SolutionTo find the number of elements in the sample space, use the formula for thenumber of combinations of 41 elements taken six at a time.

If a person buys only one ticket, the probability of winning is


Random Selection

The numbers of colleges and universities in various regions of the United States in 2003 are shown in Figure 9.11. One institution is selected at random. What is the probability that the institution is in one of the three southern regions?(Source: National Center for Education Statistics)

SolutionFrom the figure, the total number of colleges and universities is 4163. Becausethere are colleges and universities in the threesouthern regions, the probability that the institution is in one of these regions is

FIGURE 9.11


West NorthCentral

441

East NorthCentral

630

East South Central284

West South Central386

MiddleAtlantic

624South

Atlantic700

NewEngland

261Pacific

563

Mountain274


�1370

4163� 0.329.

700 � 284 � 386 � 1370

�1

4,496,388 .P�E� �

n�E�n�S�

� 4,496,388

�41 � 40 � 39 � 38 � 37 � 36

6 � 5 � 4 � 3 � 2 � 1

n�S� � 41C6


Example 5

Example 6

333202_0907.qxd 12/5/05 11:41 AM Page 704

Mutually Exclusive EventsTwo events and (from the same sample space) are mutually exclusive if and have no outcomes in common. In the terminology of sets, the intersectionof and is the empty set, which is written as

For instance, if two dice are tossed, the event of rolling a total of 6 and theevent of rolling a total of 9 are mutually exclusive. To find the probability thatone or the other of two mutually exclusive events will occur, you can add theirindividual probabilities.

The Probability of a Union of Events

One card is selected from a standard deck of 52 playing cards. What is theprobability that the card is either a heart or a face card?

SolutionBecause the deck has 13 hearts, the probability of selecting a heart (event ) is

Similarly, because the deck has 12 face cards, the probability of selecting a facecard (event ) is

Because three of the cards are hearts and face cards (see Figure 9.12), it followsthat

Finally, applying the formula for the probability of the union of two events, youcan conclude that the probability of selecting a heart or a face card is


� 0.423.�22

52 �

13

52�

12

52�

3

52

P�A � B� � P�A� � P�B� � P�A � B�

P�A � B� �3

52.

P�B� �12

52.

B

P�A� �13

52.

A

BA

P�A � B� � 0.

BAB

ABA


Example 7

Probability of the Union of Two EventsIf and are events in the same sample space, the probability of or occurring is given by

If and are mutually exclusive, then

P�A � B� � P�A� � P�B�.

BA

P�A � B� � P�A� � P�B� � P�A � B�.

BABA

A♥

J♥Q♥

K♥ K♣

Q♣J♣

K♦

Q♦J♦

K♠

Q♠J♠

3♥2♥

4♥6♥

8♥5♥7♥

9♥10♥

Hearts

Face cards

n(A ∩ B) = 3

FIGURE 9.12

333202_0907.qxd 12/5/05 11:41 AM Page 705


Probability of Mutually Exclusive Events

The personnel department of a company has compiled data on the numbers ofemployees who have been with the company for various periods of time. Theresults are shown in the table.

If an employee is chosen at random, what is the probability that the employee has(a) 4 or fewer years of service and (b) 9 or fewer years of service?

Solutiona. To begin, add the number of employees to find that the total is 529. Next, let

event represent choosing an employee with 0 to 4 years of service. Then theprobability of choosing an employee who has 4 or fewer years of service is

b. Let event represent choosing an employee with 5 to 9 years of service. Then

Because event from part (a) and event have no outcomes in common, youcan conclude that these two events are mutually exclusive and that

So, the probability of choosing an employee who has 9 or fewer years ofservice is about 0.465.


� 0.465.

�246

529

�157

529�

89

529

P�A � B� � P�A� � P�B�

BA

P�B� �89

529 .

B

P�A� �157

529� 0.297.

A

Years of service Number of employees

0–4 157

5–9 89

10–14 74

15–19 63

20–24 42

25–29 38

30–34 37

35–39 21

40–44 8

Example 8

333202_0907.qxd 12/5/05 11:41 AM Page 706

Independent EventsTwo events are independent if the occurrence of one has no effect on the occur-rence of the other. For instance, rolling a total of 12 with two six-sided dice hasno effect on the outcome of future rolls of the dice. To find the probability thattwo independent events will occur, multiply the probabilities of each.

Probability of Independent Events

A random number generator on a computer selects three integers from 1 to 20.What is the probability that all three numbers are less than or equal to 5?

SolutionThe probability of selecting a number from 1 to 5 is

So, the probability that all three numbers are less than or equal to 5 is


Probability of Independent Events

In 2004, approximately 20% of the adult population of the United States got theirnews from the Internet every day. In a survey, 10 people were chosen at randomfrom the adult population. What is the probability that all 10 got their news fromthe Internet every day? (Source: The Gallup Poll)

SolutionLet represent choosing an adult who gets the news from the Internet every day.The probability of choosing an adult who got his or her news from the Internetevery day is 0.20, the probability of choosing a second adult who got his or hernews from the Internet every day is 0.20, and so on. Because these events areindependent, you can conclude that the probability that all 10 people got theirnews from the Internet every day is


�P�A��10 � �0.20�10 � 0.0000001.

A

�1

64.

P�A� � P�A� � P�A� � 1

41

41

4

�1

4.P�A� �

5

20


Probability of Independent EventsIf and are independent events, the probability that both and willoccur is

P�A and B� � P�A� � P�B�.

BABA

Example 9

Example 10

333202_0907.qxd 12/5/05 11:41 AM Page 707

The Complement of an EventThe complement of an event is the collection of all outcomes in the samplespace that are not in The complement of event is denoted by Because

or and because and are mutually exclusive, it follows thatSo, the probability of is

For instance, if the probability of winning a certain game is

the probability of losing the game is

Finding the Probability of a Complement

A manufacturer has determined that a machine averages one faulty unit for every1000 it produces. What is the probability that an order of 200 units will have oneor more faulty units?

SolutionTo solve this problem as stated, you would need to find the probabilities ofhaving exactly one faulty unit, exactly two faulty units, exactly three faulty units,and so on. However, using complements, you can simply find the probability thatall units are perfect and then subtract this value from 1. Because the probabilitythat any given unit is perfect is 999/1000, the probability that all 200 units areperfect is

So, the probability that at least one unit is faulty is


� 0.181

� 1 � 0.819.

P�A� � � 1 � P�A�

� 0.819.

P�A� � 999

1000200

�3

4.

P�A� � � 1 �1

4

P�A� �1

4

P�A� � � 1 � P�A�.

A�P�A� � P�A�� 1.A�AA� � � 1P�A

A�.AA.A


Probability of a ComplementLet be an event and let be its complement. If the probability of is

the probability of the complement is

P�A� � � 1 � P�A�.

P�A�,AA�A

Example 11

You are in a class with 22 otherpeople. What is the probabilitythat at least two out of the 23people will have a birthday onthe same day of the year?

The complement of theprobability that at least twopeople have the same birthdayis the probability that all 23birthdays are different. So, firstfind the probability that all 23people have different birthdaysand then find the complement.

Now, determine the proba-bility that in a room with 50people at least two people havethe same birthday.

Exploration

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In Exercises 1–6, determine the sample space for theexperiment.

1. A coin and a six-sided die are tossed.

2. A six-sided die is tossed twice and the sum of the points isrecorded.

3. A taste tester has to rank three varieties of yogurt, A, B,and C, according to preference.

4. Two marbles are selected from a bag containing two redmarbles, two blue marbles, and one yellow marble. Thecolor of each marble is recorded.

5. Two county supervisors are selected from five supervisors,A, B, C, D, and E, to study a recycling plan.

6. A sales representative makes presentations about a productin three homes per day. In each home, there may be a sale(denote by S) or there may be no sale (denote by F).

Tossing a Coin In Exercises 7–10, find the probability forthe experiment of tossing a coin three times. Use the sample space

7. The probability of getting exactly one tail

8. The probability of getting a head on the first toss

9. The probability of getting at least one head

10. The probability of getting at least two heads

Drawing a Card In Exercises 11–14, find the probabilityfor the experiment of selecting one card from a standarddeck of 52 playing cards.

11. The card is a face card.

12. The card is not a face card.

13. The card is a red face card.

14. The card is a 6 or lower. (Aces are low.)

Tossing a Die In Exercises 15–20, find the probability forthe experiment of tossing a six-sided die twice.

15. The sum is 4. 16. The sum is at least 7.

17. The sum is less than 11. 18. The sum is 2, 3, or 12.

19. The sum is odd and no more than 7.

20. The sum is odd or prime.

Drawing Marbles In Exercises 21–24, find the probabilityfor the experiment of drawing two marbles (withoutreplacement) from a bag containing one green, two yellow,and three red marbles.

21. Both marbles are red.

22. Both marbles are yellow.

23. Neither marble is yellow.

24. The marbles are of different colors.

T TH, T T T }.S � {HHH, HHT, HTH, H T T, THH, TH T,

Exercises 9.7

VOCABULARY CHECK:

In Exercises 1–7, fill in the blanks.

1. An ________ is an event whose result is uncertain, and the possible results of the event are called ________.

2. The set of all possible outcomes of an experiment is called the ________ ________.

3. To determine the ________ of an event, you can use the formula where is the number of

outcomes in the event and is the number of outcomes in the sample space.

4. If then is an ________ event, and if then is a ________ event.

5. If two events from the same sample space have no outcomes in common, then the two events are ________ ________.

6. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________.

7. The ________ of an event is the collection of all outcomes in the sample space that are not in

8. Match the probability formula with the correct probability name.

(a) Probability of the union of two events (i)

(b) Probability of mutually exclusive events (ii)

(c) Probability of independent events (ii)

(d) Probability of a complement (iv)


P�A and B� � P�A� � P�B�P�A � B� � P�A� � P�B� � P�A � B�P�A� � � 1 � P�A�P�A � B� � P�A� � P�B�

A.A

EP�E� � 1,EP�E� � 0,

n�S�

n�E�P�E� �n�E�n�S� ,

333202_0907.qxd 12/5/05 11:41 AM Page 709

www.Eduspace.com

In Exercises 25–28, you are given the probability that anevent will happen. Find the probability that the event willnot happen.

25. 26.

27.

28.

In Exercises 29–32, you are given the probability that anevent will not happen. Find the probability that the eventwill happen.

29.

30.

31.

32.

33. Data Analysis A study of the effectiveness of a fluvaccine was conducted with a sample of 500 people. Someparticipants in the study were given no vaccine, some weregiven one injection, and some were given two injections.The results of the study are listed in the table.

A person is selected at random from the sample. Find thespecified probability.

(a) The person had two injections.

(b) The person did not get the flu.

(c) The person got the flu and had one injection.

34. Data Analysis One hundred college students were inter-viewed to determine their political party affiliations andwhether they favored a balanced-budget amendment to theConstitution. The results of the study are listed in the table,where represents Democrat and represents Republican.

A person is selected at random from the sample. Find theprobability that the described person is selected.

(a) A person who doesn’t favor the amendment

(b) A Republican

(c) A Democrat who favors the amendment

35. Graphical Reasoning The figure shows the results of arecent survey in which 1011 adults were asked to gradeU.S. public schools. (Source: Phi Delta Kappa/GallupPoll)

(a) Estimate the number of adults who gave U.S. publicschools a B.

(b) An adult is selected at random. What is the probabiltythat the adult will give the U.S. public schools an A?

(c) An adult is selected at random. What is the probabiltythe adult will give the U.S. public schools a C or a D?

36. Graphical Reasoning The figure shows the results of asurvey in which auto racing fans listed their favorite typeof racing. (Source: ESPN Sports Poll/TNS Sports)

(a) What is the probability that an auto racing fan selectedat random lists NASCAR racing as his or her favoritetype of racing?

(b) What is the probability that an auto racing fan selectedat random lists Formula One or motorcycle racing ashis or her favorite type of racing?

(c) What is the probability that an auto racing fan selectedat random does not list NHRA drag racing as his or herfavorite type of racing?

Favorite Type of Racing

NASCAR59%

NHRAdrag racing

13%

Motorcycle 11%

Other 11%

Formula One6%

A 2%

Grading Public Schools

Fail 3%

C 52%

B 24%

D 12%

Don’t know7%

RD

P�E�� 61

100

P�E�� 1735

P�E�� 0.92

P�E�� 0.14

P�E� �23

P�E� �14

P�E� � 0.36P�E� � 0.7


No One Twovaccine injection injections Total

Flu 7 2 13 22

No flu 149 52 277 478

Total 156 54 290 500

Favor Not Favor Unsure Total

D 23 25 7 55

R 32 9 4 45

Total 55 34 11 100

333202_0907.qxd 12/5/05 11:41 AM Page 710


37. Alumni Association A college sends a survey to selectedmembers of the class of 2006. Of the 1254 people whograduated that year, 672 are women, of whom 124 went onto graduate school. Of the 582 male graduates, 198 went onto graduate school. An alumni member is selected atrandom. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attendgraduate school?

38. Education In a high school graduating class of 202students, 95 are on the honor roll. Of these, 71 are going onto college, and of the other 107 students, 53 are going onto college. A student is selected at random from the class.What are the probabilities that the person chosen is (a)going to college, (b) not going to college, and (c) on thehonor roll, but not going to college?

39. Winning an Election Taylor, Moore, and Jenkins arecandidates for public office. It is estimated that Moore andJenkins have about the same probability of winning, andTaylor is believed to be twice as likely to win as either ofthe others. Find the probability of each candidate winningthe election.

40. Winning an Election Three people have been nominatedfor president of a class. From a poll, it is estimated that thefirst candidate has a 37% chance of winning and thesecond candidate has a 44% chance of winning. What isthe probability that the third candidate will win?

In Exercises 41–52, the sample spaces are large and youshould use the counting principles discussed in Section9.6.

41. Preparing for a Test A class is given a list of 20 studyproblems, from which 10 will be part of an upcomingexam. A student knows how to solve 15 of the problems.Find the probabilities that the student will be able toanswer (a) all 10 questions on the exam, (b) exactly eightquestions on the exam, and (c) at least nine questions onthe exam.

42. Payroll Mix-Up Five paychecks and envelopes areaddressed to five different people. The paychecks arerandomly inserted into the envelopes. What are theprobabilities that (a) exactly one paycheck will be insertedin the correct envelope and (b) at least one paycheck willbe inserted in the correct envelope?

43. Game Show On a game show, you are given five digits toarrange in the proper order to form the price of a car. If youare correct, you win the car. What is the probability ofwinning, given the following conditions?

(a) You guess the position of each digit.

(b) You know the first digit and guess the positions of theother digits.

44. Card Game The deck of a card game is made up of 108cards. Twenty-five each are red, yellow, blue, and green,and eight are wild cards. Each player is randomly dealt aseven-card hand.

(a) What is the probability that a hand will contain exactlytwo wild cards?

(b) What is the probability that a hand will contain twowild cards, two red cards, and three blue cards?

45. Drawing a Card One card is selected at random from anordinary deck of 52 playing cards. Find the probabilitiesthat (a) the card is an even-numbered card, (b) the card is aheart or a diamond, and (c) the card is a nine or a face card.

46. Poker Hand Five cards are drawn from an ordinary deckof 52 playing cards. What is the probability that the handdrawn is a full house? (A full house is a hand that consistsof two of one kind and three of another kind.)

47. Defective Units A shipment of 12 microwave ovenscontains three defective units. A vending company hasordered four of these units, and because each is identicallypackaged, the selection will be random. What are theprobabilities that (a) all four units are good, (b) exactly twounits are good, and (c) at least two units are good?

48. Random Number Generator Two integers from 1through 40 are chosen by a random number generator.What are the probabilities that (a) the numbers are botheven, (b) one number is even and one is odd, (c) bothnumbers are less than 30, and (d) the same number ischosen twice?

49. Flexible Work Hours In a survey, people were asked ifthey would prefer to work flexible hours—even if it meantslower career advancement—so they could spend moretime with their families. The results of the survey areshown in the figure. Three people from the survey werechosen at random. What is the probability that all threepeople would prefer flexible work hours?

Rigid hours13%

Don’t know9%Flexible

hours78%

Flexible Work Hours

333202_0907.qxd 12/5/05 11:41 AM Page 711

50. Consumer Awareness Suppose that the methods used byshoppers to pay for merchandise are as shown in the circlegraph. Two shoppers are chosen at random. What is theprobability that both shoppers paid for their purchases onlyin cash?

51. Backup System A space vehicle has an independentbackup system for one of its communication networks. Theprobability that either system will function satisfactorilyduring a flight is 0.985. What are the probabilities that during a given flight (a) both systems function satisfactorily, (b) at least one system functions satisfactorily, and (c) both systems fail?

52. Backup Vehicle A fire company keeps two rescue vehi-cles. Because of the demand on the vehicles and the chanceof mechanical failure, the probability that a specific vehicleis available when needed is 90%. The availability of onevehicle is independent of the availability of the other. Findthe probabilities that (a) both vehicles are available at agiven time, (b) neither vehicle is available at a given time,and (c) at least one vehicle is available at a given time.

53. A Boy or a Girl? Assume that the probability of the birthof a child of a particular sex is 50%. In a family with fourchildren, what are the probabilities that (a) all the childrenare boys, (b) all the children are the same sex, and (c) thereis at least one boy?

54. Geometry You and a friend agree to meet at your favoritefast-food restaurant between 5:00 and 6:00 P.M. The onewho arrives first will wait 15 minutes for the other, andthen will leave (see figure). What is the probability that thetwo of you will actually meet, assuming that your arrivaltimes are random within the hour?

Your arrival time(in minutes past 5:00 P.M.)

You

r fr

iend

’s a

rriv

al ti

me

(in

min

utes

pas

t 5:0

0 P.

M.)

15 30 45 60

15

30

45

60 You meetYou meetYou don’t meet

You ar

rive f

irst

Your f

riend

arriv

es fi

rst

Half cash, half credit30%

Mostly credit7%

Only credit4%

Only cash32%

Mostly cash27%

How Shoppers Pay for Merchandise


55. Roulette American roulette is a game in which awheel turns on a spindle and is divided into 38 pockets.Thirty-six of the pockets are numbered 1–36, of whichhalf are red and half are black. Two of the pockets aregreen and are numbered 0 and 00 (see figure). Thedealer spins the wheel and a small ball in oppositedirections. As the ball slows to a stop, it has an equalprobability of landing in any of the numbered pockets.

(a) Find the probability of landing in the number 00pocket.

(b) Find the probability of landing in a red pocket.

(c) Find the probability of landing in a green pocket ora black pocket.

(d) Find the probability of landing in the number 14pocket on two consecutive spins.

(e) Find the probability of landing in a red pocket onthree consecutive spins.

(f ) European roulette does not contain the 00 pocket.Repeat parts (a)–(e) for European roulette. How dothe probabilities for European roulette comparewith the probabilities for American roulette?

298

31

633

435

2

282611

20

17

2215

24

13

0010 12

19

18

2116

23

14

0930

7

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534

336

1

2725

Model It

333202_0907.qxd 12/5/05 11:41 AM Page 712


56. Estimating A coin of diameter is dropped onto apaper that contains a grid of squares units on a side (seefigure).

(a) Find the probability that the coin covers a vertex of oneof the squares on the grid.

(b) Perform the experiment 100 times and use the resultsto approximate

Synthesis


57. If and are independent events with nonzero probabili-ties, then can occur when occurs.

58. Rolling a number less than 3 on a normal six-sided die hasa probability of . The complement of this event is to roll anumber greater than 3, and its probability is

59. Pattern Recognition and Exploration Consider a groupof people.

(a) Explain why the following pattern gives the probabili-ties that the people have distinct birthdays.

(b) Use the pattern in part (a) to write an expression for theprobability that people have distinct birthdays.

(c) Let be the probability that the people have distinctbirthdays. Verify that this probability can be obtainedrecursively by

and

(d) Explain why gives the probability that atleast two people in a group of people have the samebirthday.

(e) Use the results of parts (c) and (d) to complete thetable.

(f) How many people must be in a group so that the prob-ability of at least two of them having the same birthdayis greater than Explain.

60. Think About It A weather forecast indicates that theprobability of rain is 40%. What does this mean?

Skills Review

In Exercises 61–70, find all real solutions of the equation.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

In Exercises 71–74, sketch the graph of the solution set ofthe system of inequalities.

71.

72.

73.

74. �x2 � y2 ≤ 4x � y ≥ �2

�x2 � y ≥ �2y ≥ x � 4

� x ≤ 3y ≤ 6

5x � 2y ≥ 10

� y ≥ �3x ≥ �1

�x � y ≥ �8

2x

�5

x � 2�

�13x2 � 2x

3x � 2

�x

x � 2� 1

32x � 3

� 4 ��1

2x � 3

2x � 5

� 4

32x

� 2x

12x

� �3

x 5 � x 3 � 2x � 0

x 3 � x2 � 3x � 0

4x2 � 6x � 12 � 0

6x2 � 8 � 0

12?

nQn � 1 � Pn

Pn �365 � �n � 1�

365Pn�1.P1 � 1

nPn

n � 4

n � 3: 365

365�

364

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363

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333202_0907.qxd 12/5/05 11:41 AM Page 713


Chapter Summary9

What did you learn?

Section 9.1 Review Exercises� Use sequence notation to write the terms of sequences (p. 642). 1–8

� Use factorial notation (p. 644). 9–12

� Use summation notation to write sums (p. 646). 13–20

� Find the sums of infinite series (p. 647). 21–24

� Use sequences and series to model and solve real-life problems (p. 648). 25, 26

Section 9.2� Recognize, write, and find the nth terms of arithmetic sequences (p. 653). 27–40

� Find nth partial sums of arithmetic sequences (p. 656). 41–46

� Use arithmetic sequences to model and solve real-life problems (p. 657). 47, 48

Section 9.3� Recognize, write, and find the nth terms of geometric sequences (p. 663). 49–60

� Find nth partial sums of geometric sequences (p. 666). 61–70

� Find sums of infinite geometric series (p. 667). 71–76

� Use geometric sequences to model and solve real-life problems (p. 668). 77, 78

Section 9.4� Use mathematical induction to prove statements involving a positive integer n (p. 673). 79–82

� Recognize patterns and write the nth term of a sequence (p. 677). 83–86

� Find the sums of powers of integers (p. 679). 87–90

� Find finite differences of sequences (p. 680). 91–94

Section 9.5� Use the Binomial Theorem to calculate binomial coefficients (p. 683). 95–98

� Use Pascal’s Triangle to calculate binomial coefficients (p. 685). 99–102

� Use binomial coefficients to write binomial expansions (p. 686). 103–108

Section 9.6� Solve simple counting problems (p. 691). 109, 110

� Use the Fundamental Counting Principle to solve counting problems (p. 692). 111, 112

� Use permutations to solve counting problems (p. 693). 113, 114

� Use combinations to solve counting problems (p. 696). 115, 116

Section 9.7� Find the probabilities of events (p. 701). 117, 118

� Find the probabilities of mutually exclusive events (p. 705). 119, 120

� Find the probabilities of independent events (p. 707). 121, 122

� Find the probability of the complement of an event (p. 708). 123, 124

333202_090R.qxd 12/5/05 11:43 AM Page 714

Review Exercises 715

In Exercises 1–4, write the first five terms of thesequence. (Assume that begins with 1.)

1.

2.

3.

4.

In Exercises 5–8, write an expression for the apparent thterm of the sequence. (Assume that begins with 1.)

5.

6.

7. 8.

In Exercises 9–12, simplify the factorial expression.

9. 5! 10.

11. 12.

In Exercises 13–18, find the sum.

13. 14.

15. 16.

17. 18.

In Exercises 19 and 20, use sigma notation to write the sum.

19.

20.

In Exercises 21–24, find the sum of the infinite series.

21. 22.

23. 24.

25. Compound Interest A deposit of $10,000 is made in anaccount that earns 8% interest compounded monthly. Thebalance in the account after months is given by

(a) Write the first 10 terms of this sequence.

(b) Find the balance in this account after 10 years by find-ing the 120th term of the sequence.

26. Education The enrollment (in thousands) in HeadStart programs in the United States from 1994 to 2002 canbe approximated by the model

where is the year, with corresponding to 1994.Find the terms of this finite sequence. Use a graphing util-ity to construct a bar graph that represents the sequence.(Source: U.S. Administration for Children and Families)

In Exercises 27–30, determine whether the sequenceis arithmetic. If so, find the common difference.

27. 5, 3, 1, . . . 28. 0, 1, 3, 6, 10, . . .

29. 1, 2, . . . 30. . . .

In Exercises 31–34, write the first five terms of the arith-metic sequence.

31. 32.

33.

34.

In Exercises 35–40, find a formula for for the arithmeticsequence.

35. 36.

37. 38.

39. 40.

In Exercises 41–44, find the partial sum.

41. 42.

43. 44.

45. Find the sum of the first 100 positive multiples of 5.

46. Find the sum of the integers from 20 to 80 (inclusive).

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d � �3a1 � 25,d � 12a1 � 7,

an

a1 � 4.2, ak�1 � ak � 0.4

a1 � 25, ak�1 � ak � 3

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333202_090R.qxd 12/5/05 11:43 AM Page 715

47. Job Offer The starting salary for an accountant is $34,000with a guaranteed salary increase of $2250 per year.Determine (a) the salary during the fifth year and (b) thetotal compensation through 5 full years of employment.

48. Baling Hay In the first two trips baling hay around alarge field, a farmer obtains 123 bales and 112 bales,respectively. Because each round gets shorter, the farmerestimates that the same pattern will continue. Estimate thetotal number of bales made if the farmer takes another sixtrips around the field.

In Exercises 49–52, determine whether the sequenceis geometric. If so, find the common ratio.

49. 50.

51. 52.

In Exercises 53–56, write the first five terms of the geometricsequence.

53. 54.

55. 56.

In Exercises 57–60, write an expression for the th term ofthe geometric sequence. Then find the 20th term of thesequence.

57. 58.

59. 60.

In Exercises 61–66, find the sum of the finite geometricsequence.

61. 62.

63. 64.

65. 66.

In Exercises 67–70, use a graphing utility to find the sum ofthe finite geometric sequence.

67. 68.

69. 70.

In Exercises 71–76, find the sum of the infinite geometricseries.

71. 72.

73. 74.

75. 76.

77. Depreciation A paper manufacturer buys a machine for$120,000. During the next 5 years, it will depreciate at arate of 30% per year. (That is, at the end of each year thedepreciated value will be 70% of what it was at the begin-ning of the year.)

(a) Find the formula for the th term of a geometricsequence that gives the value of the machine full yearsafter it was purchased.

(b) Find the depreciated value of the machine after 5 fullyears.

78. Annuity You deposit $200 in an account at the beginningof each month for 10 years. The account pays 6%compounded monthly. What will your balance be at theend of 10 years? What would the balance be if the interestwere compounded continuously?

In Exercises 79–82, use mathematical induction toprove the formula for every positive integer

79.

80.

81.

82.

In Exercises 83–86, find a formula for the sum of the first terms of the sequence.

83. 9, 13, 17, 21, . . . 84. 68, 60, 52, 44, . . .

85. 1, . . . 86. 12, . . .

In Exercises 87–90, find the sum using the formulas for thesums of powers of integers.

87. 88.

89. 90.

In Exercises 91–94, write the first five terms of the sequencebeginning with the given term. Then calculate the first andsecond differences of the sequence. State whether thesequence has a linear model, a quadratic model, or neither.

91. 92.

93. 94.

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333202_090R.qxd 12/5/05 11:43 AM Page 716

Review Exercises 717

In Exercises 95–98, use the Binomial Theorem tocalculate the binomial coefficient.

95. 96.

97. 98.

In Exercises 99–102, use Pascal’s Triangle to calculate thebinomial coefficient.

99. 100.

101. 102.

In Exercises 103–108, use the Binomial Theorem to expandand simplify the expression. (Remember that )

103.

104.

105.

106.

107.

108.

109. Numbers in a Hat Slips of paper numbered 1 through14 are placed in a hat. In how many ways can you drawtwo numbers with replacement that total 12?

110. Home Theater Systems A customer in an electronicsstore can choose one of six speaker systems, one of fiveDVD players, and one of six plasma televisions to designa home theater system. How many systems can bedesigned?

111. Telephone Numbers The same three-digit prefix is usedfor all of the telephone numbers in a small town. How many different telephone numbers are possible bychanging only the last four digits?

112. Course Schedule A college student is preparing acourse schedule for the next semester. The student mayselect one of three mathematics courses, one of fourscience courses, and one of six history courses. Howmany schedules are possible?

113. Bike Race There are 10 bicyclists entered in a race. Inhow many different ways could the top three places bedecided?

114. Jury Selection A group of potential jurors has beennarrowed down to 32 people. In how many ways can ajury of 12 people be selected?

115. Apparel You have eight different suits to choose from totake on a trip. How many combinations of three suitscould you take on your trip?

116. Menu Choices A local sub shop offers five differentbreads, seven different meats, three different cheeses, andsix different vegetables. Find the total number of combi-nations of sandwiches possible.

117. Apparel A man has five pairs of socks, of which no twopairs are the same color. He randomly selects two socksfrom a drawer. What is the probability that he gets amatched pair?

118. Bookshelf Order A child returns a five-volume set ofbooks to a bookshelf. The child is not able to read, and socannot distinguish one volume from another. What is theprobability that the books are shelved in the correctorder?

119. Students by Class At a particular university, the num-bers of students in the four classes are broken down bypercents, as shown in the table.

A single student is picked randomly by lottery for a cashscholarship. What is the probability that the scholarshipwinner is

(a) a junior or senior?

(b) a freshman, sophomore, or junior?

120. Data Analysis A sample of college students, faculty,and administration were asked whether they favored aproposed increase in the annual activity fee to enhancestudent life on campus. The results of the study are listedin the table.

A person is selected at random from the sample. Findeach specified probability.

(a) The person is not in favor of the proposal.

(b) The person is a student.

(c) The person is a faculty member and is in favor of theproposal.

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9.5

9.6

9.7

Class Percent

Freshmen 31

Sophomores 26

Juniors 25

Seniors 18

Students Faculty Admin. Total

Favor 237 37 18 292

Oppose 163 38 7 208

Total 400 75 25 500

333202_090R.qxd 12/5/05 11:43 AM Page 717

121. Tossing a Die A six-sided die is tossed three times.What is the probability of getting a 6 on each roll?

122. Tossing a Die A six-sided die is tossed six times. Whatis the probability that each side appears exactly once?

123. Drawing a Card You randomly select a card from a 52-card deck. What is the probability that the card is nota club?

124. Tossing a Coin Find the probability of obtaining at leastone tail when a coin is tossed five times.

Synthesis

True or False? In Exercises 125–129, determine whetherthe statement is true or false. Justify your answer.

125.

126.

127.

128.

129. The value of is always greater than the value of

130. Think About It An infinite sequence is a function. Whatis the domain of the function?

131. Think About It How do the two sequences differ?

(a)

(b)

132. Graphical Reasoning The graphs of two sequences areshown below. Identify each sequence as arithmetic orgeometric. Explain your reasoning.

(a) (b)

133. Writing Explain what is meant by a recursion formula.

134. Writing Explain why the terms of a geometric sequencedecrease when

Graphical Reasoning In Exercises 135–138, match thesequence or sum of a sequence with its graph withoutdoing any calculations. Explain your reasoning. [Thegraphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

135.

136.

137.

138.

139. Population Growth Consider an idealized populationwith the characteristic that each member of the populationproduces one offspring at the end of every time period. Ifeach member has a life span of three time periods and thepopulation begins with 10 newborn members, then thefollowing table shows the population during the first fivetime periods.

The sequence for the total population has the property that

Find the total population during the next five time periods.

140. The probability of an event must be a real number in whatinterval? Is the interval open or closed?

n > 3.Sn � Sn�1 � Sn�2 � Sn�3,

an � �n

k�14��1

2�k�1

an � �n

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40

60

80

100

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8

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Time Period

Age Bracket 1 2 3 4 5

0–1 10 10 20 40 70

1–2 10 10 20 40

2–3 10 10 20

Total 10 20 40 70 130

333202_090R.qxd 12/5/05 11:43 AM Page 718

Chapter Test 719

Chapter Test9

Take this test as you would take a test in class. When you are finished, check yourwork against the answers given in the back of the book.

1. Write the first five terms of the sequence (Assume that begins with 1.)

2. Write an expression for the th term of the sequence.

3. Find the next three terms of the series. Then find the fifth partial sum of the series.

4. The fifth term of an arithmetic sequence is 5.4, and the 12th term is 11.0. Find the th term.

5. Write the first five terms of the sequence (Assume that begins with 1.)

In Exercises 6 –8, find the sum.

6. 7. 8.

9. Use mathematical induction to prove the formula.

10. Use the Binomial Theorem to expand the expression

11. Find the coefficient of the term in the expansion of

In Exercises 12 and 13, evaluate each expression.

12. (a) (b)

13. (a) (b)

14. How many distinct license plates can be issued consisting of one letter followed by athree-digit number?

15. Eight people are going for a ride in a boat that seats eight people. The owner of theboat will drive, and only three of the remaining people are willing to ride in the twobow seats. How many seating arrangements are possible?

16. You attend a karaoke night and hope to hear your favorite song. The karaoke songbook has 300 different songs (your favorite song is among the 300 songs). Assumingthat the singers are equally likely to pick any song and no song is repeated, what is theprobability that your favorite song is one of the 20 that you hear that night?

17. You are with seven of your friends at a party. Names of all of the 60 guests are placedin a hat and drawn randomly to award eight door prizes. Each guest is limited to oneprize. What is the probability that you and your friends win all eight of the prizes?

18. The weather report calls for a 75% chance of snow. According to this report, what isthe probability that it will not snow?

66C411C4

70 P39 P2

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2

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7

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,

n

nan ��1�n

3n � 2.

333202_090R.qxd 12/5/05 11:43 AM Page 719


Cumulative Test for Chapters 7–99

Take this test to review the material from earlier chapters. When you are finished,check your work against the answers given in the back of the book.

In Exercises 1– 4, solve the system by the specified method.

1. Substitution 2. Elimination

3. Elimination 4. Gauss-Jordan Elimination

In Exercises 5 and 6, sketch the graph of the solution set of the system of inequalities.

5. 6.

7. Sketch the region determined by the constraints. Then find the minimum andmaximum values, and where they occur, of the objective function subject to the indicated constraints.

8. A custom-blend bird seed is to be mixed from seed mixtures costing $0.75 per poundand $1.25 per pound. How many pounds of each seed mixture are used to make 200pounds of custom-blend bird seed costing $0.95 per pound?

9. Find the equation of the parabola passing through the points and

In Exercises 10 and 11, use the system of equations at the left.

10. Write the augmented matrix corresponding to the system of equations.

11. Solve the system using the matrix found in Exercise 10 and Gauss-Jordan elimination.

In Exercises 12–15, use the following matrices to find each of the following, if possible.

12. 13.

14. 15.

16. Find the determinant of the matrix at the left.

17. Find the inverse of the matrix (if it exists): �13

�5

27

�7

�1�10�15�.

ABA � 2B

�2BA � B

B � [�11

30]A � [ 4

�102],

�6, 4�.�3, 1�,�0, 4�,y � ax2 � bx � c

x � 4y ≤ 202x � y ≤ 12

x ≥ 0y ≥ 0

z � 3x � 2y,

� x � y >

5x � 2y <

6

10�2x � y ≥x � 3y ≤

�3

2

� x � 3y � 2z �

�2x � y � z �

4x � y � z �

�7�5

3��2x � 4y � z �

x � 2y � 2z �

x � 3y � z �

3�6

1

� x � 3y �

2x � 4y �

�1

0� y �

2�y � 2� �

3 � x2

x � 1

SYSTEM FOR 10 AND 11

��x

2x

3x

� 2y

� y

� 3y

� z �

� 2z �

� 4z �

9

�9

7

MATRIX FOR 16

�81

�2

036

�5�1

4�

333202_090R.qxd 12/8/05 10:55 AM Page 720

Cumulative Test for Chapters 7–9 721

18. The percents (by age group) of the total amounts spent on three types of footwear ina recent year are shown in the matrix. The total amounts (in millions) spent by eachage group on the three types of footwear were $442.20 (14–17 age group),$466.57(18–24 age group), and $1088.09 (25–34 age group). How many dollars worthof gym shoes, jogging shoes, and walking shoes were sold that year? (Source:National Sporting Goods Association)

In Exercises 19 and 20, use Cramer’s Rule to solve the system of equations.

19.

20.

21. Find the area of the triangle shown in the figure.

22. Write the first five terms of the sequence (assume that begins with 1).

23. Write an expression for the th term of the sequence.

24. Find the sum of the first 20 terms of the arithmetic sequence

25. The sixth term of an arithmetic sequence is 20.6, and the ninth term is 30.2.

(a) Find the 20th term.

(b) Find the th term.

26. Write the first five terms of the sequence (assume that begins with 1).

27. Find the sum:

28. Use mathematical induction to prove the formula

29. Use the Binomial Theorem to expand and simplify

In Exercises 30–33, evaluate the expression.

30. 31. 32. 33.

In Exercises 34 and 35, find the number of distinguishable permutations of thegroup of letters.

34. B, A, S, K, E, T, B, A, L, L 35. A, N, T, A, R, C, T, I, C, A

36. A personnel manager at a department store has 10 applicants to fill three differentsales positions. In how many ways can this be done, assuming that all the applicantsare qualified for any of the three positions?

37. On a game show, the digits 3, 4, and 5 must be arranged in the proper order to formthe price of an appliance. If the digits are arranged correctly, the contestant wins theappliance. What is the probability of winning if the contestant knows that the price isat least $400?

10C3�84�25P27P3

�z � 3�4.

3 � 7 � 11 � 15 � . . . � �4n � 1� � n�2n � 1�.

��

i�0 1.3� 1

10�i�1.

nan � 3�2�n�1

n

8, 12, 16, 20, . . . .

6!8

, . . .5!7

,4!6

,3!5

,2!4

,

n

nan ��1�n�1

2n � 3

�5x � 4y � 3z �

�3x � 8y � 7z �

7x � 5y � 6z �

7

�9

�53

�8x � 3y �

3x � 5y �

�52

5

x1 3

6

−1−2

1

2 4

2

5

( 2, 3)−

(1, 5)

(4, 1)

y

FIGURE FOR 21

MATRIX FOR 18

Gym Jogging Walkingshoes shoes shoes

�0.090.060.12

0.090.100.25

0.030.050.12�Age

group �

14�17

18�24

25�34

333202_090R.qxd 12/8/05 10:56 AM Page 721

ProofEach of these properties follows directly from the properties of real numbers.

1. n terms

The Distributive Property is used in the proof of Property 2.

2.

The proof of Property 3 uses the Commutative and Associative Properties ofAddition.

3.

The proof of Property 4 uses the Commutative and Associative Properties ofAddition and the Distributive Property.

4.

� �n

i�1 ai � �

n

i�1 bi

� �a1 � a2 � a3 � . . . � an � � �b1 � b2 � b3 � . . . � bn�

� �a1 � a2 � a3 � . . . � an � � ��b1 � b2 � b3 � . . . � bn�

�n

i�1 �ai � bi� � �a1 � b1� � �a2 � b2� � �a3 � b3� � . . . � �an � bn�

� �n

i�1 ai � �

n

i�1 bi

� �a1 � a2 � a3 � . . . � an � � �b1 � b2 � b3 � . . . � bn�

�n

i�1 �ai � bi� � �a1 � b1� � �a2 � b2� � �a3 � b3� � . . . � �an � bn�

� c�n

i�1 ai � c�a1 � a2 � a3 � . . . � an�

�n

i�1

cai � ca1 � ca2 � ca3 � . . . � can

�n

i�1

c � c � c � c � . . . � c � cn

722

Proofs in Mathematics

Properties of Sums (p. 647)

1. is a constant.

2. is a constant.

3.

4. �n

i�1

�ai � bi� � �n

i�1

ai � �n

i�1

bi

�n

i�1

�ai � bi� � �n

i�1

ai � �n

i�1

bi

c�n

i�1

cai � c�n

i�1

ai ,

c�n

i�1

c � cn,

Infinite SeriesThe study of infinite series wasconsidered a novelty in the four-teenth century. Logician RichardSuiseth, whose nickname wasCalculator, solved this problem.

If throughout the first half of agiven time interval a variation con-tinues at a certain intensity;throughout the next quarter of theinterval at double the intensity;throughout the following eighthat triple the intensity and so adinfinitum; The average intensity forthe whole interval will be theintensity of the variation during thesecond subinterval (or double theintensity).This is the same as saying thatthe sum of the infinite series

is 2.

12

�24

�38

� . . . �n2n � . . .

333202_090R.qxd 12/5/05 11:43 AM Page 722

723

ProofBegin by generating the terms of the arithmetic sequence in two ways. In the firstway, repeatedly add to the first term to obtain

In the second way, repeatedly subtract from the th term to obtain

If you add these two versions of the multiples of subtract out and you obtain

n terms

Proof

Multiply by r.

Subtracting the second equation from the first yields

So, and, because you have Sn � a1�1 � rn

1 � r �.r � 1,Sn�1 � r� � a1�1 � rn�,

Sn � rSn � a1 � a1rn.

rSn � a1r � a1r2 � a1r

3 � . . . � a1rn�1 � a1r

n

Sn � a1 � a1r � a1r2 � . . . � a1r

n�2 � a1rn�1

Sn �n

2�a1 � an�.

2Sn � n�a1 � an�

2Sn � �a1 � an� � �a1 � an� � �a1 � an� � . . . � �a1 � an�

dSn,

� an � �an � d� � �an � 2d� � . . . � �an � �n � 1�d�.

Sn � an � an�1 � an�2 � . . . � a3 � a2 � a1

nd

� a1 � �a1 � d� � �a1 � 2d� � . . . � �a1 � �n � 1�d�.

Sn � a1 � a2 � a3 � . . . � an�2 � an�1 � an

d

The Sum of a Finite Arithmetic Sequence (p. 656)

The sum of a finite arithmetic sequence with terms is

Sn �n

2�a1 � an �.

n

The Sum of a Finite Geometric Sequence (p. 666)

The sum of the finite geometric sequence

with common ratio is given by Sn � �n

i�1

a1ri�1 � a1�1 � rn

1 � r �.r � 1

a1, a1r, a1r2, a1r

3, a1r4, . . . , a1r

n�1

333202_090R.qxd 12/5/05 11:43 AM Page 723

ProofThe Binomial Theorem can be proved quite nicely using mathematical induction.The steps are straightforward but look a little messy, so only an outline of theproof is presented.

1. If you have and the formula isvalid.

2. Assuming that the formula is true for the coefficient of is

To show that the formula is true for look at the coefficient ofin the expansion of

From the right-hand side, you can determine that the term involving is the sum of two products.

So, by mathematical induction, the Binomial Theorem is valid for all positiveintegers n.

� k�1Cr xk�1�ryr

� �k � 1�!�k � 1 � r�!r!xk�1�ry r

� k!�k � 1 � r � r��k � 1 � r�!r! xk�1�ry r

� �k � 1 � r�k!

�k � 1 � r�!r!�

k!r

�k � 1 � r�!r!xk�1�ry r

� k!

�k � r�!r!�

k!

�k � 1 � r�!�r � 1�!xk�1�ry r

� kCr xk�ry r��x� � � kCr�1xk�1�ry r�1��y�

xk�1�ryr

�x � y�k�1 � �x � y�k�x � y�.

xk�1�ryrn � k � 1,

kCr �k!

�k � r�!r!�

k�k � 1��k � 2� . . . �k � r � 1�r!

.

xk�ryrn � k,

�x � y�1 � x1 � y1 � 1C0x � 1C1y,n � 1,

724

The Binomial Theorem (p. 683)

In the expansion of

the coefficient of is

nCr �n!

�n � r�!r!.

xn�ryr

�x � y�n � xn � nxn�1y � . . . � nCr xn�ry r � . . . � nxyn�1 � yn

�x � y�n

333202_090R.qxd 12/5/05 11:43 AM Page 724

1. Let and consider the sequence given by

Use a graphing utility to compute the first 10 terms of thesequence and make a conjecture about the value of as approaches infinity.

2. Consider the sequence

(a) Use a graphing utility to graph the first 10 terms of thesequence.

(b) Use the graph from part (a) to estimate the value of as approaches infinity.

(c) Complete the table.

(d) Use the table from part (c) to determine (if possible) thevalue of as approaches infinity.

3. Consider the sequence

(a) Use a graphing utility to graph the first 10 terms of thesequence.

(b) Use the graph from part (a) to describe the behavior ofthe graph of the sequence.

(c) Complete the table.

(d) Use the table from part (c) to determine (if possible) thevalue of as approaches infinity.

4. The following operations are performed on each term of anarithmetic sequence. Determine if the resulting sequence isarithmetic, and if so, state the common difference.

(a) A constant is added to each term.

(b) Each term is multiplied by a nonzero constant

(c) Each term is squared.

5. The following sequence of perfect squares is not arithmetic.

1, 4, 9, 16, 25, 36, 49, 64, 81, . . .

However, you can form a related sequence that is arithmeticby finding the differences of consecutive terms.(a) Write the first eight terms of the related arithmetic

sequence described above. What is the th term of thissequence?

(b) Describe how you can find an arithmetic sequence thatis related to the following sequence of perfect cubes.

1, 8, 27, 64, 125, 216, 343, 512, 729, . . .

(c) Write the first seven terms of the related sequence inpart (b) and find the th term of the sequence.

(d) Describe how you can find the arithmetic sequence thatis related to the following sequence of perfect fourthpowers.

1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, . . .

(e) Write the first six terms of the related sequence in part(d) and find the th term of the sequence.

6. Can the Greek hero Achilles, running at 20 feet per second,ever catch a tortoise, starting 20 feet ahead of Achilles andrunning at 10 feet per second? The Greek mathematicianZeno said no. When Achilles runs 20 feet, the tortoise willbe 10 feet ahead. Then, when Achilles runs 10 feet, thetortoise will be 5 feet ahead. Achilles will keep cutting thedistance in half but will never catch the tortoise. The tableshows Zeno’s reasoning. From the table you can see thatboth the distances and the times required to achieve themform infinite geometric series. Using the table, show thatboth series have finite sums. What do these sums represent?

7. Recall that a fractal is a geometric figure that consists of apattern that is repeated infinitely on a smaller and smallerscale. A well-known fractal is called the SierpinskiTriangle. In the first stage, the midpoints of the three sidesare used to create the vertices of a new triangle, which isthen removed, leaving three triangles. The first three stagesare shown on the next page. Note that each remaining triangle is similar to the original triangle. Assume that thelength of each side of the original triangle is one unit.

n

n

n

C.

C

nan

an � 3 � ��1�n.

nan

nan

an �n � 1n2 � 1

.

nxn

n � 1, 2, . . .xn �12

xn�1 �1

xn�1,

xnx0 � 1

P.S. Problem Solving

This collection of thought-provoking and challenging exercises further exploresand expands upon concepts learned in this chapter.

n 1 10 100 1000 10,000

an

n 1 10 101 1000 10,001

an

Distance (in feet) Time (in seconds)

20 1

10 0.5

5 0.25

2.5 0.125

1.25 0.0625

0.625 0.03125

725

333202_090R.qxd 12/5/05 11:43 AM Page 725

Write a formula that describes the side length of the trianglesthat will be generated in the th stage. Write a formula for thearea of the triangles that will be generated in the th stage.

FIGURE FOR 7

8. You can define a sequence using a piecewise formula. Thefollowing is an example of a piecewise-defined sequence.

(a) Write the first 10 terms of the sequence.

(b) Choose three different values for other thanFor each value of find the first 10 terms of

the sequence. What conclusions can you make aboutthe behavior of this sequence?

9. The numbers 1, 5, 12, 22, 35, 51, are calledpentagonal numbers because they represent the numbers ofdots used to make pentagons, as shown below. Use mathe-matical induction to prove that the th pentagonal number

is given by

10. What conclusion can be drawn from the following infor-mation about the sequence of statements

(a) is true and implies

(b) are all true.

(c) are all true, but the truth of does notimply that is true.

(d) is true and implies

11. Let be the Fibonacci sequence.

(a) Use mathematical induction to prove that

(b) Find the sum of the first 20 terms of the Fibonaccisequence.

12. The odds in favor of an event occurring are the ratio of theprobability that the event will occur to the probability thatthe event will not occur. The reciprocal of this ratiorepresents the odds against the event occurring.

(a) Six marbles in a bag are red. The odds against choosinga red marble are 4 to 1. How many marbles are in thebag?

(b) A bag contains three blue marbles and sevenyellow marbles. What are the odds in favor of choosinga blue marble? What are the odds against choosing ablue marble?

(c) Write a formula for converting the odds in favor of anevent to the probability of the event.

(d) Write a formula for converting the probability of anevent to the odds in favor of the event.

13. You are taking a test that contains only multiple choicequestions (there are five choices for each question). You areon the last question and you know that the answer is not Bor D, but you are not sure about answers A, C, and E. Whatis the probability that you will get the right answer if youtake a guess?

14. A dart is thrown at the circular target shown below. Thedart is equally likely to hit any point inside the target. Whatis the probability that it hits the region outside the triangle?

15. An event has possible outcomes, which have the valuesThe probabilities of the outcomes occur-

ring are The expected value of an eventis the sum of the products of the outcomes’ probabilities

and their values,

(a) To win California’s Super Lotto Plus game, you mustmatch five different numbers chosen from the numbers1 to 47, plus one Mega number chosen from the num-bers 1 to 27. You purchase a ticket for $1. If the jack-pot for the next drawing is $12,000,000, what is theexpected value for the ticket?

(b) You are playing a dice game in which you need toscore 60 points to win. On each turn, you roll two six-sided dice. Your score for the turn is 0 if the dice do notshow the same number, and the product of the numberson the dice if they do show the same number. What isthe expected value for each turn? How many turns willit take on average to score 60 points?

V � p1x1 � p2x2 � . . . � pnxn.A

Vp1, p2, . . ., pn.nx1, x2, . . ., xn.

nA

6

f1 � f2 � . . . � fn � fn�2 � 1.

f1, f2, . . . , fn, . . .

P2k�2.P2kP2

Pk�1

PkP1, P2, and P3

P1, P2, P3, . . . , P50

Pk�1.PkP3

Pn?

Pn �n�3n � 1�

2.

Pn

n

. . .

a1,a1 � 7�.�a1

an � �an�1

2,

3an�1 � 1,

if an�1 is even

if an�1 is odda1 � 7,

nn

726

333202_090R.qxd 12/5/05 11:43 AM Page 726

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