9.1-9.39.1-9.3 Sequences and Series. Quick Review

9.1-9.39.1-9.39.1-9.39.1-9.3

Sequences and SeriesSequences and Series

Quick Review

1

10

1

1 9

Evaluate each expression when 3, 2, 4 and 2.

1. ( 1)

2.

Find .

13.

4. 2 3

5. 3 and 10

n

k

k

k

k k

a r n d

a n d

a r

a

ka

ka

a a a

Quick Review Solutions

1

10

1

1 9

9

24

Evaluate each expression when 3, 2, 4 and 2.

1. ( 1)

2.

Find .

13.

4. 2 3

11

1039,36

5. 3 and 10

6

1 3

n

k

k

k

k k

a r n d

a n d

a r

a

ka

ka

a a a

What you’ll learn about• Infinite Sequences• Limits of Infinite Sequences• Arithmetic and Geometric Sequences• Sequences and Graphing Calculators

… and whyInfinite sequences, especially those with finite

limits, are involved in some key concepts of calculus.

Limit of a Sequence Let be a sequence of real numbers, and consider lim .

If the limit is a finite number , the sequence and

is the . If the limit is infinite or nonexistent,

the se

n nna a

L L

converges

limit of the sequence

quence .diverges

Example Finding Limits of Sequences

Determine whether the sequence converges or diverges. If it converges,

give the limit.

2 1 2 22,1, , , ,..., ,...

3 2 5 n

Example Finding Limits of Sequences

Determine whether the sequence converges or diverges. If it converges,

give the limit.

2 1 2 22,1, , , ,..., ,...

3 2 5 n

2lim 0, so the sequence converges to a limit of 0.n n

Arithmetic Sequence

A sequence is an if it can be written in the

form , , 2 ,..., ( 1) ,... for some constant .

The number is called the .

Each term in an arithmetic seque

na

a a d a d a n d d

d

arithmetic sequence

common difference

1

nce can be obtained recursively from

its preceding term by adding : (for all 2).n n

d a a d n

Example Arithmetic Sequences

Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the

nth term, and (d) an explicit rule for the nth term.

-2, 1, 4, 7, …

Example Arithmetic Sequences

Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the

nth term, and (d) an explicit rule for the nth term.

-2, 1, 4, 7, …

10

1 1

(a) The common difference is 3.

(b) 2 (10 1)3 25

(c) 2 3 for all 2

(d) 2 3( 1) 3 5n n

n

a

a a a n

a n n

Geometric Sequence

2 1

A sequence is a if it can be written in the

form , , ,..., ,... for some nonzero constant .

The number is called the .

Each term in a geometric sequence

n

n

a

a a r a r a r r

r

geometric sequence

common ratio

1

can be obtained recursively from

its preceding term by multiplying by : (for all 2).n n

r a a r n

Example Defining Geometric Sequences

Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

2, 6, 18,…

Example Defining Geometric Sequences

Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

2, 6, 18,…

10 1

10

1 1

1

(a) The ratio is 3.

(b) 2 3 39,366

(c) 2 and 3 for 2.

(d) 2 3 .n n

n

n

a

a a a n

a

The Fibonacci Sequence

1

2

2 1

The Fibonacci sequences can be defined recursively by

1

1

for all positive integers 3.n n n

a

a

a a a

n

9.1-9.39.1-9.39.1-9.39.1-9.3

Sequences and Series (cont.)Sequences and Series (cont.)

Quick Review

10

1

3

10

1

3

is an arithmetic sequence. Use the given information to find .

1. 5; 4

2. 5; 2

is a geometric sequence. Use the given information to find .

3. 5; 4

4. 5; 4

5. Find the sum of

n

n

a a

a d

a d

a a

a r

a r

2 the first 3 terms of the sequence .n


10

1

3

10

1

3

is an arithmetic sequence. Use the given information to find .

1. 5; 4

2. 5; 2

is a geometric sequence. Use the given information to find .

3.

41

19

1,310,7205; 4

4.

n

n

a a

a d

a d

a a

a r

a

2

5; 4

5. Find the sum of the first 3 terms of the sequence

81,920

14.

r

n

What you’ll learn about• Summation Notation• Sums of Arithmetic and Geometric

Sequences• Infinite Series• Convergences of Geometric Series

… and whyInfinite series are at the heart of integral

calculus.

Summation Notation 1 2

1

In , the sum of the terms of the sequence , ,...,

is denoted which is read "the sum of from 1 to ."

The variable is called the .

n

n

k kk

a a a

a a k n

k

summation notation

index of summation

Sum of a Finite Arithmetic Sequence

1 2

1 21

1

1

Let , ,..., be a finite arithmetic sequence with common difference .

Then the sum of the terms of the sequence is

...

2

2 ( 1)2

n

n

k nk

n

a a a d

a a a a

a an

na n d

Example Summing the Terms of an Arithmetic Sequence

A corner section of a stadium has 6 seats along the front row. Each

successive row has 3 more seats than the row preceding it. If the top

row has 24 seats, how many seats are in the entire section?

Example Summing the Terms of an Arithmetic

Sequence

A corner section of a stadium has 6 seats along the front row. Each

successive row has 3 more seats than the row preceding it. If the top

row has 24 seats, how many seats are in the entire section?

1

1

The number of seats in the rows form an arithmetic sequence with

6, 24, and 3. Solving

( 1)

24 6 3( 1)

7

Apply the Sum of a Finite Sequence Theorem:

6 24Sum of chairs 7 105. T

2

n

n

a a d

a a n d

n

n

here are 105 seats in the section.

Sum of a Finite Geometric Sequence

1 2

1 21

1

Let , ,..., be a finite geometric sequence with common ratio .

Then the sum of the terms of the sequence is

...

1

1

n

n

k nk

n

a a a r

a a a a

a r

r

Infinite Series

1 21

An infinite series is an expression of the form

... ...k n

ka a a a

Sum of an Infinite Geometric Series

1

1The geometric series converges if and only if | | 1.

If it does converge, the sum is .1

k

ka r r

a

r

Example Summing Infinite Geometric Series

1

1

Determine whether the series converges. If it converges, give the sum.

2 0.25k

k

Example Summing Infinite Geometric Series

1

1

Determine whether the series converges. If it converges, give the sum.

2 0.25k

k

Since | | 0.25 1, the series converges.

2 8The sum is .

1 1 0.25 3

r

a

r

9.49.49.49.4

Mathematical InductionMathematical Induction

Quick Review

2

3 2

2

1. Expand the product ( 2)( 4).

Factor the polynomial.

2. 7 10

3. 3 3 1

4. Find ( ) given ( ) .1

5. Find ( 1) given ( ) 1.

k k k

n n

n n n

xf t f x

xf t f x x


2

3

3 2

22

32

1. Expand the product ( 2)( 4).

Factor the polynomial.

2. 7 10

3. 3 3 1

4. Find ( ) given ( ) . 1

5. Find ( 1) given ( ) 1.

6 8

2 5

1

12 2

k k k

n n

n n n

xf t f x

xf t f x

k k k

n n

n

t

txtt

What you’ll learn about• The Tower of Hanoi Problem• Principle of Mathematical Induction• Induction and Deduction

… and whyThe principle of mathematical induction is a valuable technique for proving

combinatorial formulas.

The Tower of Hanoi Solution

The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2n – 1.

Principle of Mathematical Induction

Let Pn be a statement about the integer n. Then Pn is true for all positive integers n provided the following conditions are satisfied:

1. (the anchor) P1 is true;

2. (inductive step) if Pk is true, then Pk+1 is true.

9.59.59.59.5

The Binomial TheoremThe Binomial Theorem

Quick Review

2

2

2

2

3

Use the distributive property to expand the binomial.

1.

2. ( 2 )

3. (2 3 )

4. (2 )

5.

x y

a b

c d

x y

x y


2

2

2

2

3

2 2

2 2

2 2

2 2

3 2 2 3

Use the distributive property to expand the binomial.

1.

2. ( 2 )

3. (2 3 )

2

4 4

4

4. (2 )

5

12 9

4 4

. 3 3

x y

a b

c d

x

x xy y

a ab b

c cd d

x xy y

x x y y yy x

y

x

What you’ll learn about• Powers of Binomials• Pascal’s Triangle• The Binomial Theorem• Factorial Identities

… and whyThe Binomial Theorem is a marvelous study in combinatorial patterns.

Binomial Coefficient

The binomial coefficients that appear in the expansion of ( )

are the values of for 0,1, 2,3,..., .

A classical notation for , especially in the context of binomial

coefficients, is .

n

n r

n r

a b

C r n

C

n

r

Both notations are read " choose ."n r

Example Using nCr to Expand a Binomial

4

Expand , using a calculator to compute the binomial coefficients.a b

Example Using nCr to Expand a Binomial

4

Expand , using a calculator to compute the binomial coefficients.a b

44 3 2 2 3

Enter 4 0,1,2,3,4 into the calculator to find the binomial

coefficients for 4. The calculator returns the list 1,4,6,4,1 .

Using these coefficients, construct the expansion:

4 6 4

n rC

n

a b a a b a b ab

4 .b

The Binomial Theorem

1

For any positive integer ,

... ... ,0 1

!where .

!( )!

nn n n r r n

n r

n

n n n na b a a b a b b

r n

n nC

r r n r

Basic Factorial Identities

For any integer 1, ! 1 !

For any integer 0, 1 ! 1 !

n n n n

n n n n

9.69.69.69.6

Counting PrinciplesCounting Principles

Quick Review

Give the number of objects described.

1. The number of cards in a standard deck.

2. The number of face cards in a standard deck.

3. The number of vertices of a octogon.

4. The number of faces on a cubical die.

5. The number of possible totals when two dice are rolled.

Quick Review Solutions Give the number of objects described.

1. The number of cards in a standard deck.

2. The number of face cards in a standard deck.

3. The number of vertices of a octogon.

4. The number of fac

52

12

8

es on a cubical die.

5. The number of possible totals when two dice are rolled

6

. 11

What you’ll learn about• Discrete Versus Continuous• The Importance of Counting• The Multiplication Principle of Counting• Permutations• Combinations• Subsets of an n-Set

… and whyCounting large sets is easy if you know the correct

formula.

Multiplication Principle of Counting

1 2

1 1

2 2

If a procedure has a sequence of stages , ,..., and if

can occur in ways,

can occur in ways

can occur in ways,

then the number of ways that the procedure can occur is the

produ

n

n n

P S S S

S r

S r

S r

P

1 2ct ... .

nrr r

Example Using the Multiplication

PrincipleIf a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

Example Using the Multiplication

PrincipleIf a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.

Permutations of an n-Set

There are n! permutations of an n-set.

Example Distinguishable Permutations

Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

Example Distinguishable Permutations

Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.

Distinguishable Permutations

1 2

1 2

There are ! distinguishable permutations of an -set containing

distinguishable objects.

If an -set contains objects of a first kind, objects of a second

kind, and so on, with ... , tk

n n n

n n n

n n n n

1 2 3

hen the number of

!distinguishable permutations of the -set is .

! ! ! !k

nn

n n n n

Permutations Counting Formula

The number of permutations of objects taken at a time is

!denoted and is given by for 0 .

!

If , then 0.

n r n r

n r

n r

nP P r n

n r

r n P

Combination Counting Formula

The number of combinations of objects taken at a time is

!denoted and is given by for 0 .

! !

If , then 0.

n r n r

n r

n r

nC C r n

r n r

r n C

Example Counting Combinations

How many 10 person committees can be formed from a group of 20 people?

Example Counting Combinations

How many 10 person committees can be formed from a group of 20 people?

20 10

Notice that order is not important. Using combinations,

20!184,756.

10! 20 10 !

There are 184,756 possible committees.

C

Formula for Counting Subsets of an n-Set

There are 2 subsets of a set with objects (including the

empty set and the entire set).

n n

9.79.79.79.7

ProbabilityProbability

Quick Review

How many outcomes are possible for the following experiments.

1. Two coins are tossed.

2. Two different 6-sided dice are rolled.

3. Two chips are drawn simultaneously without replacement from

a jar with 8

4 2

8 2

chips.

4. Two different cards are drawn from a standard deck of 52.

5. Evaluate without using a calculator. C

C


How many outcomes are possible for the following experiments.

1. Two coins are tossed.

2. Two different 6-sided dice are rolled.

3. Two chips are drawn simultaneously without replacement from

a

4

36

jar

4 2

8 2

with 8 chips.

4. Two different cards are drawn from a standard deck of 52.

5. Evaluate without using a calculat

28

1326

or. 14 3/C

C

What you’ll learn about• Sample Spaces and Probability Functions• Determining Probabilities• Venn Diagrams and Tree Diagrams• Conditional Probability• Binomial Distributions

… and whyEveryone should know how mathematical the

“laws of chance” really are.

Probability of an Event (Equally Likely Outcomes)

If is an event in a finite, nonempty sample space of equally likely

outcomes, then the of the event is

the number of outcomes in ( ) .

the number of outcomes in

E S

E

EP E

S

probability

Probability Distribution for the Sum of Two Fair Dice

Outcome Probability2 1/363 2/364 3/365 4/366 5/367 6/368 5/369 4/3610 3/3611 2/3612 1/36

Example Rolling the Dice

Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

Example Rolling the Dice

Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

The event consists of the outcomes 4,8,12 . To get the probability

of we add up the probabilities of the outcomes in :

3 5 1 9 1( ) .

36 36 36 36 4

E

E E

P E

Probability Function

A is a function that assigns a real number

to each outcome in a sample space subject to the following conditions:

1. 0 ( ) 1;

2. the sum of the probabilities of all outcomes in

P

S

P O

probability function

is 1;

3. ( ) 0.

S

P

Probability of an Event (Outcomes not Equally

Likely)

Let be a finite, nonempty sample space in which every outcome

has a probability assigned to it by a probability function . If is

any event in , the of the event is the sum of the

pro

S

P E

S Eprobability

babilities of all the outcomes contained in . E

Strategy for Determining Probabilities

1. Determine the sample space of all possible outcomes. When possible,

choose outcomes that are equally likely.

2. If the sample space has equally likely outcomes, the probability of an

event is determEthe number of outcomes in

ined by counting: ( ) .the number of outcomes in

3. If the sample space does not have equally likely outcomes, determine

the probability function. (This is not always easy

EP E

S

to do.) Check to be sure

that the conditions of a probability function are satisfied. Then the

probability of an event is determined by adding up the probabilities

of all the outcomes contained in .

E

E

Example Choosing Chocolates

Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?



12 3

The experiment in question is the selection of three chocolates,

without regard to order, from a box of 12. There are 220

outcomes of this experiment. The event E consists of all possible

combinat

C

5 3

ions of 3 that can be chosen, without regard to order, from

the 5 vanilla cremes available. There are 10 ways.

Therefore, ( ) 10 / 220 1/ 22.

C

P E

Multiplication Principle of Probability

Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2.





The probability of picking a vanilla creme on the first draw is 5/12.

Under the assumption that a vanilla creme was selected in the first

draw, the probability of picking a vanilla creme on the second draw is

4/11. Under the assumption that a vanilla creme was selected in the first

and second draw, the probability of picking a vanilla creme on the third

draw is 3/10. By the Multiplication Principle, the probability of picking

5 4 3 60 1a vanilla creme on all three picks is .

12 11 10 1320 22

Conditional Probability Formula

( and )If the event depends on the event , then ( | ) .

( )

P A BB A P B A

P A

Binomial DistributionSuppose an experiment consists of -independent repetitions of an

experiment with two outcomes, called "success" and "failure." Let

(success) and (failure) . (Note that 1 .)

Then the terms in th

n

P p P q q p e binomial expansion of ( ) give the respective

probabilities of exactly , 1,..., 2, 1, 0 successes.

np q

n n

n

Number of successes out of Probability

independent repetitionsn

1

1 1

1

n

n

p

nn p q

n

1

0

n

n

npq

r

q

Example Shooting Free Throws

Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?


Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

15

(15 successes) 0.92 0.286P


Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?


Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

10 515(10 successes)= 0.92 0.08 0.00427

10P

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9.1-9.39.1-9.3 Sequences and Series. Quick Review