838 Chapter 12 Sequences and Series

Lessons 12.4–12.5

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MIXED REVIEW of Problem SolvingMIXED REVIEW of Problem Solving

1. MULTI-STEP PROBLEM A ball is dropped from a height of 12 feet. Each time the ball hits the ground, it bounces to 70% of its previous height.

a. Write an infinite series to model the total distance traveled by the ball, excluding the distance traveled before the first bounce.

b. Find the total distance traveled by the ball, including the distance traveled before the first bounce.

2. MULTI-STEP PROBLEM A fractal tree starts with a single branch (the trunk). At each stage, the new branches from the previous stage each grow two more branches as shown.

Stage 1 Stage 2 Stage 3

. . .

a. List the number of new branches in each of the first six stages.

b. Is the sequence of numbers from part (a) arithmetic, geometric, or neither?

c. Write an explicit rule and a recursive rule for the sequence from part (a).

3. GRIDDED ANSWER What is the sum of the first three iterates of the function f(x) 5 x2 2 8 when the initial value is x0 5 2?

4. OPEN-ENDED Give an example of an explicit rule for a sequence and a recursive rule for the same sequence.

5. SHORT RESPONSE Why does the sum of an infinite geometric series not exist if !r! " 1 where r is the common ratio?

6. SHORT RESPONSE The length l1 of the first loop of a spring is 16 inches. The length l2 of the second loop is 0.9 times the length of the first loop. The length l3 of the third loop is 0.9 times the length of the second loop, and so on. If the spring could have infinitely many loops, would its length be finite or infinite? Explain. If its length is finite, find the length.

7. EXTENDED RESPONSE You take out a five year loan of $10,000 to buy a car. The loan has an annual interest rate of 6.5% compounded monthly. Each month you make a monthly payment of $196 (except the last month when you make a payment of only $165).

a. Find the monthly interest rate. Then write a recursive rule for the amount of money you owe after n months.

b. How much money do you owe after 12 months?

c. Suppose you had decided to pay an additional $50 with each monthly payment. Use a graphing calculator to find the number of months you would have needed to repay the loan.

d. In your opinion, is it beneficial to pay the additional $50 with each payment? Explain your reasoning.

8. GRIDDED ANSWER A tree farm initially has 8000 trees. Each year 10% of the trees are harvested and 500 seedlings are planted. What number of trees eventually exists on the farm after an extended period of time?

9. OPEN-ENDED Write an infinite geometric series that has a sum of 4.

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Chapter Summary 839

BIG IDEAS For Your NotebookAnalyze Sequences

The information below highlights the similarities and differences between arithmetic and geometric sequences.

Arithmetic Sequence Geometric Sequence

n1

1

an

an 5 a1 1 (n 2 1)d

First term: a1

Commondifference: d

Graph islinear. n1

1

an

an 5 a1r n 2 1

First term: a1

Common ratio: r

Graph is exponential.

Find Sums of Series

The most common formulas for sums of series are shown below.

Arithmetic Series Geometric Series Infi nite Geometric Series

Sum of the first n terms:

Sn 5 n 1 a1 1 an } 2 2

Example:

4 1 9 1 14 1 19 1 24

S5 5 5 1 4 1 24 } 2 2 5 70

Sum of the first n terms:

Sn 5 a1 1 1 2 r n }

1 2 r 2 , r ! 1

Example:

3 1 6 1 12 1 24

S4 5 3 1 1 2 24 }

1 2 2 2 5 45

Sum of the series:

S 5 a1 }

1 2 r , !r! < 1

Example:

5 1 1 1 0.2 1 0.04 1 . . .

S 5 5 } 1 2 0.2

5 6.25

Other common sum formulas:

i 5 1

" n

1 5 n i 5 1

" n

i 5 n(n 1 1) } 2

i 5 1

" n

i2 5 n(n 1 1)(2n 1 1) } 6

Use Recursive Rules

The table shows explicit and recursive rules for arithmetic and geometric sequences.

Explicit Rule Recursive Rule

Arithmetic Sequence

Example: 3, 5, 7, 9, 11, . . .

an 5 a1 1 (n 2 1)d

an 5 1 1 2n

an 5 an 2 1 1 d

a1 5 3, an 5 an 2 1 1 2

Geometric Sequence

Example: 8, 4, 2, 1, 0.5, . . .

an 5 a1r n 2 1

an 5 8(0.5)n 2 1

an 5 r p an 2 1

a1 5 8, an 5 0.5an 2 1

12Big Idea 1

Big Idea 2

Big Idea 3

CHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARYCHAPTER SUMMARY

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840 Chapter 12 Sequences and Series

CHAPTER REVIEW

REVIEW KEY VOCABULARY

REVIEW EXAMPLES AND EXERCISESUse the review examples and exercises below to check your understanding of the concepts you have learned in each lesson of Chapter 12.

EXAMPLES 5 and 6on p. 797for Exs. 5–8

VOCABULARY EXERCISES

1. Copy and complete: The values in the range of a sequence are called the ? of the sequence.

2. WRITING How can you determine whether a sequence is arithmetic?

3. Copy and complete: A(n) ? rule gives an as a function of the term’s position number n in the sequence.

4. Copy and complete: In a(n) ? sequence, the ratio of any term to the previous term is constant.

• sequence, p. 794

• terms of a sequence, p. 794

• series, p. 796

• summation notation, p. 796

• sigma notation, p. 796

• arithmetic sequence, p. 802

• common difference, p. 802

• arithmetic series, p. 804

• geometric sequence, p. 810

• common ratio, p. 810

• geometric series, p. 812

• partial sum, p. 820

• explicit rule, p. 827

• recursive rule, p. 827

• iteration, p. 830

Define and Use Sequences and Series pp. 794–800

E X A M P L E

Find the sum of the series i 5 1

! 4

(i2 2 4).

a1 5 12 2 4 5 23 First term

a2 5 22 2 4 5 0 Second term

a3 5 32 2 4 5 5 Third term

a4 5 42 2 4 5 12 Fourth term

The sum of the series is i 5 1

! 4

(i2 2 4) 5 23 1 0 1 5 1 12 5 14.

EXERCISES

Find the sum of the series.

5. n 5 1

! 6

(n2 1 7) 6. i 5 2

! 6

(10 2 4i) 7. i 5 1

! 17

i 8. k 5 1

! 25

k2

12.1

CHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEW12 classzone.com• Multi-Language Glossary• Vocabulary practice

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Chapter Review 841

EXAMPLES2, 3, 4, and 5on pp. 803–805for Exs. 9–16

Analyze Arithmetic Sequences and Series pp. 802–809

E X A M P L E

Write a rule for the nth term of the sequence 9, 13, 17, 21, 25, . . . .

The sequence is arithmetic with first term a1 5 9 and common difference d 5 4. So, a rule for the nth term is:

an 5 a1 1 (n 2 1)d Write general rule.

5 9 1 (n 2 1)(4) Substitute 9 for a1 and 4 for d.

5 5 1 4n Simplify.

EXERCISES

Write a rule for the nth term of the arithmetic sequence.

9. 8, 5, 2, 21, 24, . . . 10. d 5 7, a8 5 54 11. a4 5 27, a11 5 69

Find the sum of the series.

12. i 5 1

! 15

(3 1 2i) 13. i 5 1

! 26

(25 2 3i) 14. i 5 1

! 22

(6i 2 5) 15. i 5 1

! 30

(284 1 8i)

16. COMPUTER Joe buys a $600 computer on layaway by making a $200 down payment and then paying $25 per month. Write a rule for the total amount of money paid on the computer after n months.

12.2

EXAMPLES 2, 3, 4, and 5on pp. 811–813for Exs. 17–23

Analyze Geometric Sequences and Series pp. 810–817

E X A M P L E

Find the sum of the series i 5 1

! 7

5(3)i 2 1.

The series is geometric with first term a1 5 5 and common ratio r 5 3.

S7 5 a1 1 1 2 r7 }

1 2 r 2 Write rule for S7.

5 5 1 1 2 37 }

1 2 3 2 Substitute 5 for a1 and 3 for r.

5 5465 Simplify.

EXERCISES

Write a rule for the nth term of the geometric sequence.

17. 256, 64, 16, 4, 1, . . . 18. r 5 5, a2 5 200 19. a1 5 144, a3 5 16

Find the sum of the series.

20. i 5 1

! 6

3(5)i 2 1 21. i 5 1

! 9

8(2)i 2 1 22. i 5 1

! 5

15 1 2 } 3

2 i 2 1 23.

i 5 1 !

7

40 1 1 } 2

2 i 2 1

12.3

classzone.comChapter Review Practice

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842 Chapter 12 Sequences and Series

EXAMPLES 2 and 5on pp. 821–822for Exs. 24–31

12Find Sums of Infinite Geometric Series pp. 820–825

E X A M P L E

Find the sum of the series i 5 1

! `

1 4 } 5 2 i 2 1, if it exists.

For this series, a1 5 1 and r 5 4 } 5

. Because "r" < 1, the sum of this series exists.

The sum is S 5 a1 }

1 2 r 5 1 }

1 2 4 } 5

5 5.

EXERCISES

Find the sum of the infinite geometric series, if it exists.

24. i 5 1

! `

3 1 5 } 8

2 i 2 1 25.

i 5 1 !

`

7 1 2 3 } 4

2 i 2 1 26.

i 5 1 !

`

4(1.3)i 2 1 27. i 5 1

! `

20.2(0.5)i 2 1

Write the repeating decimal as a fraction in lowest terms.

28. 0.888. . . 29. 0.546546546. . . 30. 0.3787878. . . 31. 0.7838383. . .

12.4

CHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEWCHAPTER REVIEW

EXAMPLES 1, 2, and 3on pp. 827–828for Exs. 32–38

Use Recursive Rules with Sequences and Functions pp. 827–833

E X A M P L E

Write a recursive rule for the sequence 6, 10, 14, 18, 22, . . . .

The sequence is arithmetic with first term a1 5 6 and common differenced 5 10 2 6 5 4.

an 5 an 2 1 1 d General recursive rule for an

5 an 2 1 1 4 Substitute 4 for d.

So, a recursive rule for the sequence is a1 5 6, an 5 an 2 1 1 4.

EXERCISES

Write the first five terms of the sequence.

32. a1 5 4, an 5 an 2 1 1 9 33. a1 5 8, an 5 5an 2 1 34. a1 5 2, an 5 n p an 2 1

Write a recursive rule for the sequence.

35. 6, 18, 54, 162, 486, . . . 36. 4, 6, 9, 13, 18, . . . 37. 7, 13, 19, 25, 31, . . .

38. POPULATION A town’s population increases at a rate of about 1% per year. In 2000, the town had a population of 26,000. Write a recursive rule for the town’s population Pn in year n. Let n 5 1 represent 2000.

12.5

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Chapter Test 843

CHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TESTCHAPTER TEST12Tell whether the sequence is arithmetic, geometric, or neither. Explain.

1. 5, 9, 13, 17, . . . 2. 3, 6, 12, 24, . . . 3. 40, 10, 5 } 2

, 5 } 8

, . . . 4. 4, 7, 12, 19, . . .

Write the first six terms of the sequence.

5. an 5 6 2 n2 6. an 5 7n3 7. a1 5 4 8. a1 5 21 an 5 5an 2 1 an 5 an 2 1 1 6

Write the next term of the sequence, and then write a rule for the nth term.

9. 5, 11, 17, 23, . . . 10. 3, 15, 75, 375, . . . 11. 6 } 5

, 7 } 10

, 8 } 15

, 9 } 20

, . . . 12. 1.6, 3.2, 4.8, 6.4, . . .

Find the sum of the series.

13. i 5 1

! 48

i 14. n 5 1

! 28

n2 15. i 5 1

! 10

(4i 2 9) 16. i 5 1

! 19

(2i 1 5)

17. i 5 1

! 5

9(2)i 2 1 18. i 5 1

! 6

12 1 1 } 3

2 i 2 1 19.

i 5 1 !

`

8 1 3 } 4

2 i 2 1 20.

i 5 1 !

`

20 1 3 } 10

2 i 2 1

Write the repeating decimal as a fraction in lowest terms.

21. 0.111. . . 22. 0.464646. . . 23. 0.187187187. . . 24. 0.3252525. . .

Write a recursive rule for the sequence.

25. 2, 12, 72, 432, . . . 26. 3, 10, 17, 24, . . . 27. 135, 45, 15, 5, . . . 28. 1, 23, 9, 227, . . .

Find the first three iterates of the function for the given initial value.

29. f(x) 5 3x 2 7, x0 5 4 30. f(x) 5 8 2 5x, x0 5 1 31. f(x) 5 x2 1 2, x0 5 21

32. QUILTS Use the pattern of checkerboard quilts shown.

n 5 1, an 5 1 n 5 2, an 5 2 n 5 3, an 5 5 n 5 4, an 5 8

a. What does n represent for each quilt? What does an represent?

b. Make a table that shows n and an for n 5 1, 2, 3, 4, 5, 6, 7, and 8.

c. Use the rule an 5 n2 }

2 1 1 }

4 [1 2 (21)n] to find an for n 5 1, 2, 3, 4, 5, 6, 7,

and 8. Compare these values with the results in your table. What can you conclude about the sequence defined by this rule?

33. AUDITIONS Several rounds of auditions are being held to cast the three main parts in a play. There are 3072 actors at the first round of auditions. In each successive round of auditions, one fourth of the actors from the previous round remain. Find a rule for the number an of actors in the nth round of auditions. For what values of n does your rule make sense?

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