a-bltzmc11 967-1052-hr

967

Sequences,Induction, andProbability 11

967

Something incredible has happened.

Your college roommate, a gifted athlete,

has been given a six-year contract with a

professional baseball team. He will be playing

against the likes of Alex Rodriguez and Manny

Ramirez. Management offers him three options. One

is a beginning salary of $1,700,000 with annual increases

of $70,000 per year starting in the second year. A second

option is $1,700,000 the first year with an annual increase of

2% per year beginning in the second year. The third option

involves less money the first year—$1,500,000—but there is an

annual increase of 9% yearly after that. Which option offers the

most money over the six-year contract?

A similar problem appears as Exercise 67 in Exercise Set 11.3, and thisproblem appears as the Group Exercise on page 1002.

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 967

11.1

968 Chapter 11 Sequences, Induction, and Probability

Sequences and Summation NotationSequences

Many creations in nature involve intricate mathe-matical designs, including a variety of spirals. For

example, the arrangement of the individual florets inthe head of a sunflower forms spirals. In some species,there are 21 spirals in the clockwise direction and 34 inthe counterclockwise direction. The precise numbers

depend on the species of sunflower: 21 and 34, or 34 and55, or 55 and 89, or even 89 and 144.

This observation becomes even more interestingwhen we consider a sequence of numbers investigated by

Leonardo of Pisa, also known as Fibonacci, an Italianmathematician of the thirteenth century. The Fibonaccisequence of numbers is an infinite sequence that begins asfollows:

The first two terms are 1. Every term thereafter is the sum of the two precedingterms. For example, the third term, 2, is the sum of the first and second terms:

The fourth term, 3, is the sum of the second and third terms:and so on. Did you know that the number of spirals in a daisy or a sunflower, 21 and34, are two Fibonacci numbers? The number of spirals in a pine cone, 8 and 13, anda pineapple, 8 and 13, are also Fibonacci numbers.

We can think of the Fibonacci sequence as a function.The terms of the sequence

are the range values for a function whose domain is the set of positive integers.

Thus, andso on.

The letter with a subscript is used to represent function values of asequence, rather than the usual function notation. The subscripts make up thedomain of the sequence and they identify the location of a term. Thus, repre-sents the first term of the sequence, represents the second term, the thirdterm, and so on. This notation is shown for the first six terms of the Fibonaccisequence:

The notation represents the term, or general term, of a sequence. Theentire sequence is represented by 5an6.

nthan

a1 = 1

1,

a2 = 1

1,

a3 = 2

2,

a4 = 3

3,

a5 = 5

5,

a6 = 8

8.

a3a2

a1

a

f112 = 1, f122 = 1, f132 = 2, f142 = 3, f152 = 5, f162 = 8, f172 = 13,

Range: 1, 1, 2, 3, 5, 8, 13, Á

T T T T T T T

Domain: 1, 2, 3, 4, 5, 6, 7, Á

f

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Á

1 + 2 = 3,1 + 1 = 2.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Á .

Objectives

� Find particular terms of asequence from the generalterm.

� Use recursion formulas.

� Use factorial notation.

� Use summation notation.

Sec t i on

Definition of a SequenceAn infinite sequence is a function whose domain is the set of positiveintegers.The function values, or terms, of the sequence are represented by

Sequences whose domains consist only of the first positive integers are calledfinite sequences.

n

a1 , a2 , a3 , a4 , Á , an , Á .

5an6

Fibonacci Numberson the PianoKeyboard

Numbers in the Fibonaccisequence can be found in anoctave on the piano keyboard.Theoctave contains 2 black keys in onecluster and 3 black keys in anothercluster, for a total of 5 black keys. Italso has 8 white keys, for a total of13 keys. The numbers 2, 3, 5, 8, and13 are the third through seventhterms of the Fibonacci sequence.

One Octave

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 968

Section 11.1 Sequences and Summation Notation 969

Study TipThe factor in the general termof a sequence causes the signs of theterms to alternate between positiveand negative, depending on whether

is even or odd.n

1-12n

Writing Terms of a Sequence from the General Term

Write the first four terms of the sequence whose term, or general term, is given:

a. b.

Solution

a. We need to find the first four terms of the sequence whose general term is To do so, we replace in the formula with 1, 2, 3, and 4.

The first four terms are 7, 10, 13, and 16. The sequence defined by can be written as

b. We need to find the first four terms of the sequence whose general term is

To do so, we replace each occurrence of in the formula with

1, 2, 3, and 4.

The first four terms are and The sequence defined by can be written as

Check Point 1 Write the first four terms of the sequence whose term, orgeneral term, is given:

a. b.

Although sequences are usually named with the letter any lowercase letter can be used. For example, the first four terms of the sequence are

and Because a sequence is a function whose domain is the set of positive integers,

the graph of a sequence is a set of discrete points. For example, consider thesequence whose general term is How does the graph of this sequence differan =

1n .

b4 =116 .b1 =

12 , b2 =

14 , b3 =

18 ,

5bn6 = E A12 BnF

a,

an =

1-12n

2n+ 1

.an = 2n + 5

nth

- 12

, 18

, - 126

, 180

, Á , 1-12n

3n- 1

, Á .

1-12n

3n- 1

180 .-

12 , 18 , -

126 ,

a1, 1stterm

a2, 2ndterm

a3, 3rdterm

a4, 4thterm

(–1)1

31-1=

–1

3-1=–

1

2

(–1)2

32-1=

1

9-1=

1

8

(–1)3

33-1=

–1

27-1=–

1

26

(–1)4

34-1=

1

81-1=

1

80

nan =

1-12n

3n- 1

.

7, 10, 13, 16, Á , 3n + 4, Á .

an = 3n + 4

3 � 1+4=3+4=7a1, 1stterm 3 � 2+4=6+4=10

a2, 2ndterm

3 � 3+4=9+4=13a3, 3rdterm 3 � 4+4=12+4=16

a4, 4thterm

nan = 3n + 4.

an =

1-12n

3n- 1

.an = 3n + 4

nth

EXAMPLE 1� Find particular terms of a sequence from the general term.

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 969

an

n

(2, q) (3, a) (4, ~)(1, 1)

1

2

3

1 2 3 4

Figure 11.1(b) The graph of

5an6 = e1nf

y

x

(2, q) (3, a) (4, ~)(1, 1)

1

2

3

1 2 3 4

Figure 11.1(a) The graph of

f1x2 =

1x

, x 7 0

Technology

Graphing utilities can write the termsof a sequence and graph them. Forexample, to find the first six terms

of enter

The first few terms of the sequenceare shown in the viewing rectangle. Bypressing the right arrow key to scrollright, you can see the remaining terms.

SEQ (1÷x, x, 1, 6, 1).

Generalterm

Stopat a6.

Startat a1.

Variableused ingeneralterm

The“step”from

a1 to a2,a2 to a3,etc., is 1.

5an6 = e1nf ,


from the graph of the function The graph of is shown in

Figure 11.1(a) for positive values of To obtain the graph of the sequence

remove all the points from the graph of except those whose are positive

integers. Thus, we remove all points except and so on. The

remaining points are the graph of the sequence shown in Figure 11.1(b).Notice that the horizontal axis is labeled and the vertical axis is labeled an .n

5an6 = E 1n F ,

11, 12, A2, 12 B , A3, 13 B , A4, 14 B ,

x-coordinatesf

5an6 = E 1n F ,x.

f1x2 =1xf1x2 =

1x ?

� Use recursion formulas.

Recursion FormulasIn Example 1, the formulas used for the term of a sequence expressed the term asa function of the number of the term. Sequences can also be defined using recursionformulas.A recursion formula defines the term of a sequence as a function of theprevious term. Our next example illustrates that if the first term of a sequence isknown, then the recursion formula can be used to determine the remaining terms.

Using a Recursion Formula

Find the first four terms of the sequence in which and for

Solution Let’s be sure we understand what is given.

Now let’s write the first four terms of this sequence.

This is the given first term.

Use with Thus,

Substitute 5 for

Again use with

Substitute 17 for

Notice that is defined in terms of We used with

Use the value of the third term, obtained above.

The first four terms are 5, 17, 53, and 161.

a3 , = 31532 + 2 = 161

n = 4.an = 3an - 1 + 2,a3 .a4 a4 = 3a3 + 2

a2 . = 31172 + 2 = 53

n = 3.an = 3an - 1 + 2, a3 = 3a2 + 2

a1 . = 3152 + 2 = 17

a2 = 3a2 - 1 + 2 = 3a1 + 2.n = 2.an = 3an - 1 + 2, a2 = 3a1 + 2

a1 = 5

a1=5 and an = 3an-1 +2

The firstterm is 5.

Each termafter the first

3 times theprevious term

is plus 2.

n Ú 2.an = 3an - 1 + 2a1 = 5

EXAMPLE 2

nthn,

nth

Comparing a continuous graph to the graph of a sequence

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 970

TechnologyMost calculators have factorial keys.To find 5!, most calculators use oneof the following:

Many Scientific Calculators

Many Graphing Calculators

Because becomes quite large as increases, your calculator willdisplay these larger values inscientific notation.

nn!

5� ! � � ENTER �.

5� x! �


Check Point 2 Find the first four terms of the sequence in which andfor

Factorial NotationProducts of consecutive positive integers occur quite often in sequences. Theseproducts can be expressed in a special notation, called factorial notation.

n Ú 2.an = 2an - 1 + 5a1 = 3

� Use factorial notation.

Factorials from 0through 20

As increases, grows veryrapidly. Factorial growth is moreexplosive than exponential growthdiscussed in Chapter 4.

n!n

Factorial NotationIf is a positive integer, the notation (read “ factorial”) is the product of allpositive integers from down through 1.

0! (zero factorial), by definition, is 1.

0! = 1

n! = n1n - 121n - 22Á 132122112

nnn!n

The values of for the first six positive integers are

Factorials affect only the number or variable that they follow unless groupingsymbols appear. For example,

whereas

In this sense, factorials are similar to exponents.

Finding Terms of a Sequence Involving Factorials

Write the first four terms of the sequence whose term is

Solution We need to find the first four terms of the sequence.To do so, we replace

each in with 1, 2, 3, and 4.

The first four terms are 2, 4, 4, and 83 .

a1, 1stterm

a2, 2ndterm

a3, 3rdterm

a4, 4thterm

21

(1-1)!

2

0!

2

1= = =2

22

(2-1)!

4

1!

4

1= = =4

23

(3-1)!

8

2!

8

2 � 1= = =4

24

(4-1)!

16

3!

16

3 � 2 � 1= ==

16

6

8

3=

2n

1n - 12!n

an =

2n

1n - 12!.

nth

EXAMPLE 3

12 # 32! = 6! = 6 # 5 # 4 # 3 # 2 # 1 = 720.

2 # 3! = 213 # 2 # 12 = 2 # 6 = 12

6! = 6 # 5 # 4 # 3 # 2 # 1 = 720.

5! = 5 # 4 # 3 # 2 # 1 = 120

4! = 4 # 3 # 2 # 1 = 24

3! = 3 # 2 # 1 = 6

2! = 2 # 1 = 2

1! = 1

n!

0! 11! 12! 23! 64! 245! 1206! 7207! 50408! 40,3209! 362,88010! 3,628,80011! 39,916,80012! 479,001,60013! 6,227,020,80014! 87,178,291,20015! 1,307,674,368,00016! 20,922,789,888,00017! 355,687,428,096,00018! 6,402,373,705,728,00019! 121,645,100,408,832,00020! 2,432,902,008,176,640,000

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 971


Check Point 3 Write the first four terms of the sequence whose term is

When evaluating fractions with factorials in the numerator and the denominator,try to reduce the fraction before performing the multiplications. For example, consider

Rather than write out 26! as the product of all integers from 26 down to 1, we can

express 26! as

In this way, we can divide both the numerator and the denominator by the commonfactor, 21!.

Evaluating Fractions with Factorials

Evaluate each factorial expression:

a. b.

Solution

a.

b.

Check Point 4 Evaluate each factorial expression:

a. b.

Summation NotationIt is sometimes useful to find the sum of the first terms of a sequence. For example,consider the cost of raising a child born in the United States in 2006 to a middle-income ($43,200–$72,600 per year) family, shown in Table 11.1.

n

n!1n - 12!

.14!

2!12!

1n + 12!

n!=

1n + 12 #

n!

n! = n + 1

10!2!8!

=

10 # 9 # 8!

2 # 1 # 8! =

902

= 45

1n + 12!

n!.

10!2!8!

EXAMPLE 4

26!21!

=

26 # 25 # 24 # 23 # 22 # 21!

21! = 26 # 25 # 24 # 23 # 22 = 7,893,600

26! = 26 # 25 # 24 # 23 # 22 # 21!.

26!21!

.

an =

201n + 12!

.

nth

� Use summation notation.

Table 11.1 The Cost of Raising a Child Born in the U.S. in 2006 to a Middle-Income Family

2006

$10,600

Year

Average Cost

2007

$10,930

2008

$11,270

2009

$11,960

2010

$12,330

2011

$12,710

2012

$12,950

2013

$13,350

2014

$13,760

Childis under 1.

Childis 1.

Childis 2.

Childis 3.

Childis 4.

Childis 5.

Childis 6.

Childis 7.

Childis 8.

2015

$13,970

Year

Average Cost

2016

$14,400

2017

$14,840

2018

$16,360

2019

$16,860

2020

$17,390

2021

$18,430

2022

$19,000

2023

$19,590

Childis 9.

Childis 10.

Childis 11.

Childis 12.

Childis 13.

Childis 14.

Childis 15.

Childis 16.

Childis 17.

Source: U.S. Department of Agriculture

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 972


We can let represent the cost of raising a child in year where corresponds to 2006, to 2007, to 2008, and so on.The terms of the finitesequence in Table 11.1 are given as follows:

10,600,

a1

10,930,

a2

11,270,

a3

11,960,

a4

12,330,

a5

12,710,

a6

12,950,

a7

13,350,

a8

13,760,

a9

13,970,

a10

14,400,

a11

14,840,

a12

16,360,

a13

16,860,

a14

17,390,

a15

18,430,

a16

19,000,

a17

19,590.

a18

n = 3n = 2n = 1n,an

= 260,700.

+ 13,970 + 14,400 + 14,840 + 16,360 + 16,860 + 17,390 + 18,430 + 19,000 + 19,590

= 10,600 + 10,930 + 11,270 + 11,960 + 12,330 + 12,710 + 12,950 + 13,350 + 13,760

a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18

Why might we want to add the terms of this sequence? We do this to find thetotal cost of raising a child born in 2006 from birth through age 17. Thus,

We see that the total cost of raising a child born in 2006 from birth through age 17 is$260,700.

There is a compact notation for expressing the sum of the first terms of asequence. For example, rather than write

n

a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 ,

we can use summation notation to express the sum as

We read this expression as “the sum as goes from 1 to 18 of ”The letter is called

the index of summation and is not related to the use of to represent You can think of the symbol (the uppercase Greek letter sigma) as an

instruction to add up the terms of a sequence.©

2-1.i

iai .i

a18

i = 1 ai .

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

where is the index of summation, is the upper limit of summation, and 1 is thelower limit of summation.

ni

an

i = 1ai = a1 + a2 + a3 + a4 +

Á+ an ,

n

Any letter can be used for the index of summation.The letters and are usedcommonly. Furthermore, the lower limit of summation can be an integer other than 1.

When we write out a sum that is given in summation notation, we areexpanding the summation notation. Example 5 shows how to do this.

Using Summation Notation

Expand and evaluate the sum:

a. b. c. a5

i = 13.a

7

k = 431-22k - 54a

6

i = 11i2

+ 12

EXAMPLE 5

ki, j,

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 973


Solution

a. To find we must replace in the expression with all consecutive

integers from 1 to 6, inclusive.Then we add.

b. The index of summation in is First we evaluate

for all consecutive integers from 4 through 7, inclusive. Then we add.

c. To find we observe that every term of the sum is 3. The notation

through 5 indicates that we must add the first five terms of a sequence in which every term is 3.

Check Point 5 Expand and evaluate the sum:

a. b. c.

Although the domain of a sequence is the set of positive integers, any integers can be used for the limits of summation. For a given sum, we can varythe upper and lower limits of summation, as well as the letter used for the indexof summation. By doing so, we can produce different-looking summationnotations for the same sum. For example, the sum of the squares of the first fourpositive integers, can be expressed in a number of equivalentways:

= 12+ 22

+ 32+ 42

= 30.

a5

k = 2 1k - 122 = 12 - 122 + 13 - 122 + 14 - 122 + 15 - 122

= 12+ 22

+ 32+ 42

= 30

a3

i = 0 1i + 122 = 10 + 122 + 11 + 122 + 12 + 122 + 13 + 122

a4

i = 1 i2

= 12+ 22

+ 32+ 42

= 30

12+ 22

+ 32+ 42,

a5

i = 1 4.a

5

k = 3 12k

- 32a6

i = 1 2i2

a5

i = 1 3 = 3 + 3 + 3 + 3 + 3 = 15

i = 1a5

i = 1 3,

= -100

= 11 + 1-372 + 59 + 1-1332

= 116 - 52 + 1-32 - 52 + 164 - 52 + 1-128 - 52

+ 31-226 - 54 + 31-227 - 54

a7

k = 431-22k - 54 = 31-224 - 54 + 31-225 - 54

1-22k - 5k.a7

k = 431-22k - 54

= 97 = 2 + 5 + 10 + 17 + 26 + 37

+ 152+ 12 + 162

+ 12

a6

i = 11i2

+ 12 = 112+ 12 + 122

+ 12 + 132+ 12 + 142

+ 12

i2+ 1ia

6

i = 11i2

+ 12,TechnologyGraphing utilities can calculate thesum of a sequence. For example, tofind the sum of the sequence inExample 5(a), enter

Then press 97 should bedisplayed. Use this capability to verifyExample 5(b).

� ENTER �;

� SUM � � SEQ � 1x2+ 1, x, 1, 6, 12.

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 974


Writing Sums in Summation Notation

Express each sum using summation notation:

a. b.

Solution In each case, we will use 1 as the lower limit of summation and for theindex of summation.

a. The sum has seven terms, each of the form startingat and ending at Thus,

b. The sum

has terms, each of the form starting at and ending at Thus,

Check Point 6 Express each sum using summation notation:

a. b.

Table 11.2 contains some important properties of sums expressed insummation notation.

1 +

12

+

14

+

18

+Á

+

1

2n - 1 .12+ 22

+ 32+

Á+ 92

1 +

13

+

19

+

127

+Á

+

1

3n - 1 = an

i = 1

1

3i - 1 .

i = n.i = 11

3i - 1 ,n

1 +

13

+

19

+

127

+Á

+

1

3n - 1

13+ 23

+ 33+

Á+ 73

= a7

i = 1 i3.

i = 7.i = 1i3,13

+ 23+ 33

+Á

+ 73

i

1 +

13

+

19

+

127

+Á

+

1

3n - 1 .13+ 23

+ 33+

Á+ 73

EXAMPLE 6

Table 11.2 Properties of Sums

Property Example

1. any real numberan

i = 1 cai = c a

n

i = 1 ai , c

Conclusion: a4

i = 1 3i2

= 3 a4

i = 1 i2

3 a4

i = 1 i2

= 3112+ 22

+ 32+ 422 = 3 # 12

+ 3 # 22+ 3 # 32

+ 3 # 42

a4

i = 1 3i2

= 3 # 12+ 3 # 22

+ 3 # 32+ 3 # 42

2. an

i = 1 1ai + bi2 = a

n

i = 1 ai + a

n

i = 1 bi

Conclusion: a4

i = 1 1i + i22 = a

4

i = 1 i + a

4

i = 1 i2

= 11 + 122 + 12 + 222 + 13 + 322 + 14 + 422

a4

i = 1 i + a

4

i = 1 i2

= 11 + 2 + 3 + 42 + 112+ 22

+ 32+ 422

a4

i = 1 1i + i22 = 11 + 122 + 12 + 222 + 13 + 322 + 14 + 422

3. an

i = 1 1ai - bi2 = a

n

i = 1 ai - a

n

i = 1 bi

Conclusion: a5

i = 3 1i2

- i32 = a5

i = 3 i2

- a5

i = 3 i3

= 132- 332 + 142

- 432 + 152- 532

a5

i = 3 i2

- a5

i = 3 i3

= 132+ 42

+ 522 - 133+ 43

+ 532

a5

i = 3 1i2

- i32 = 132- 332 + 142

- 432 + 152- 532

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 75


Exercise Set 11.1Practice ExercisesIn Exercises 1–12, write the first four terms of each sequence whosegeneral term is given.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

The sequences in Exercises 13–18 are defined using recursionformulas. Write the first four terms of each sequence.

13. and for

14. and for

15. and for

16. and for

17. and for

18. and for

In Exercises 19–22, the general term of a sequence is given andinvolves a factorial. Write the first four terms of each sequence.

19. 20.

21. 22.

In Exercises 23–28, evaluate each factorial expression.

23. 24.

25. 26.

27. 28.

In Exercises 29–42, find each indicated sum.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42. a5

i = 1

1i + 22!

i!a5

i = 1

i!1i - 12!

a4

i = 0

1-12i + 1

1i + 12!a4

i = 0

1-12i

i!

a7

i = 3 12a

9

i = 5 11

a4

i = 2 a -

13b

i

a4

i = 1 a -

12b

i

a4

k = 1 1k - 321k + 22a

5

k = 1 k1k + 42

a5

i = 1 i3

a4

i = 1 2i2

a6

i = 1 7ia

6

i = 1 5i

12n + 12!

12n2!

1n + 22!

n!

20!2!18!

16!2!14!

18!16!

17!15!

an = -21n - 12!an = 21n + 12!

an =

1n + 12!

n2an =

n2

n!

n Ú 2an = 3an - 1 - 1a1 = 5

n Ú 2an = 2an - 1 + 3a1 = 4

n Ú 2an = 5an - 1a1 = 2

n Ú 2an = 4an - 1a1 = 3

n Ú 2an = an - 1 + 4a1 = 12

n Ú 2an = an - 1 + 5a1 = 7

an =

1-12n + 1

2n+ 1

an =

1-12n + 1

2n- 1

an =

3n

n + 5an =

2n

n + 4

an = 1-12n + 11n + 42an = 1-12n1n + 32

an = a - 13b

n

an = 1-32n

an = a13b

n

an = 3n

an = 4n - 1an = 3n + 2

In Exercises 43–54, express each sum using summation notation.Use 1 as the lower limit of summation and for the index ofsummation.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

In Exercises 55–60, express each sum using summation notation.Use a lower limit of summation of your choice and for the indexof summation.

55.

56.

57.

58.

59.

60.

Practice PlusIn Exercises 61–68, use the graphs of and to find eachindicated sum.

5bn65an6

1a + d2 + 1a + d22 +Á

+ 1a + dn2

a + 1a + d2 + 1a + 2d2 +Á

+ 1a + nd2

a + ar + ar2+

Á+ ar14

a + ar + ar2+

Á+ ar12

6 + 8 + 10 + 12 +Á

+ 32

5 + 7 + 9 + 11 +Á

+ 31

k

a + ar + ar2+

Á+ arn - 1

1 + 3 + 5 +Á

+ 12n - 12

19

+

292 +

393 +

Á+

n

9n

4 +

42

2+

43

3+

Á+

4n

n

13

+

24

+

35

+Á

+

1616 + 2

12

+

23

+

34

+Á

+

1414 + 1

1 + 2 + 3 +Á

+ 40

1 + 2 + 3 +Á

+ 30

5 + 52+ 53

+Á

+ 512

2 + 22+ 23

+Á

+ 211

14+ 24

+ 34+

Á+ 124

12+ 22

+ 32+

Á+ 152

i

The Graph of {bn}

n

bn

1 2 3 4 5−1

12345

−2−3−4−5

The Graph of {an}

n

an

1 2 3 4 5−1

12345

−2−3−4−5

61. 62.

63. 64. a5

i = 1 1ai + 3bi2a

5

i = 1 12ai + bi2

a5

i = 1 1bi

2- 12a

5

i = 1 1ai

2+ 12

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 976


$18$16$14$12$10$8$6$4$2

United States Online Ad Spending

Year2000

8.1

2001

7.2

2002

6.1

2003

8.1

2004

10.0

2005

13.1

2006

16.7

Onl

ine

Ad

Spen

ding

(bill

ions

of d

olla

rs)

Source: eMarketer

65. 66.

67. 68.

Application Exercises69. Advertisers don’t have to fear that they’ll face a sea of “sold

out” signs as they rush to the Internet.The growing number ofpopular sites filled with user-created content, includingMySpace.com and YouTube.com, provide plenty of inventoryfor advertisers who can’t find space on top portals such asYahoo. The bar graph shows U.S. online ad spending, inbillions of dollars, from 2000 through 2006.

a5

i = 1 ai

2- a

5

i = 3 bi

2a

5

i = 1 ai

2+ a

5

i = 1 bi

2

a5

i = 4 ¢ai

bi≤3

a5

i = 4 ¢ai

bi≤2 Let represent spending for consumer drug ads, in billions of

dollars, years after 2001.

a. Use the numbers given in the graph to find and interpret

b. The finite sequence whose general term is

where models spending for consumer

drug ads, in billions of dollars, years after 2001. Use the

model to find Does this seem reasonable in terms of

the actual sum in part (a), or has model breakdownoccurred?

71. A deposit of $6000 is made in an account that earns 6%interest compounded quarterly. The balance in the accountafter quarters is given by the sequence

Find the balance in the account after five years. Round to thenearest cent.

72. A deposit of $10,000 is made in an account that earns 8%interest compounded quarterly. The balance in the accountafter quarters is given by the sequence

Find the balance in the account after six years. Round to thenearest cent.

Writing in Mathematics73. What is a sequence? Give an example with your description.

74. Explain how to write terms of a sequence if the formula forthe general term is given.

75. What does the graph of a sequence look like? How is itobtained?

76. What is a recursion formula?

77. Explain how to find if is a positive integer.

78. Explain the best way to evaluate without a calculator.

79. What is the meaning of the symbol Give an example withyour description.

80. You buy a new car for $24,000. At the end of years, thevalue of your car is given by the sequence

Find and write a sentence explaining what this valuerepresents. Describe the term of the sequence in terms ofthe value of your car at the end of each year.

Technology ExercisesIn Exercises 81–85, use a calculator’s factorial key to evaluate eachexpression.

81. 82. 83.20!300

a30020b !

200!198!

ntha5

an = 24,000a34b

n

, n = 1, 2, 3, Á .

n

©?

900!899!

nn!

an = 10,000a1 +

0.084b

n

, n = 1, 2, 3, Á .

n

an = 6000a1 +

0.064b

n

, n = 1, 2, 3, Á .

n

15a

5

i = 1 ai .

n

n = 1, 2, 3, 4, 5,

an = 0.65n + 2.3,

15a

5

i = 1 ai .

nan

Let represent online ad spending, in billions of dollars, yearsafter 1999.

a. Use the numbers given in the graph to find and interpret

b. The finite sequence whose general term is

where models online ad spending, inbillions of dollars, years after 1999. Use the model to find

Does this underestimate or overestimate the actual

sum in part (a)? By how much?

70. More and more television commercial time is devoted todrug companies as hucksters for the benefits and risks oftheir wares. The bar graph shows the amount that drugcompanies spent on consumer drug ads, in billions of dollars,from 2002 through 2006.

17a

7

i = 1 ai .

nan ,n = 1, 2, 3, Á , 7,

an = 0.5n2- 1.5n + 8,

17a

7

i = 1 ai .

nan

Spending for Consumer Drug Ads

Year2006

5.5

2005

4.9

2004

4.5

2003

3.6

2002

2.9

$6

$5

$4

$3

$2

$1Spen

ding

for

Dru

g A

ds(b

illio

ns o

f dol

lars

)

Source: Nielsen Monitor-Plus

A-BLTZMC11_967-1052-hr 8-10-2008 14:59 Page 977


84. 85.

86. Use the (sequence) capability of a graphing utility toverify the terms of the sequences you obtained for any fivesequences from Exercises 1–12 or 19–22.

87. Use the (sum of the sequence) capability of agraphing utility to verify any five of the sums you obtained inExercises 29–42.

88. As increases, the terms of the sequence

get closer and closer to the number (where ). Use acalculator to find and comparingthese terms to your calculator’s decimal approximation for

Many graphing utilities have a sequence-graphing mode that plotsthe terms of a sequence as points on a rectangular coordinate system.Consult your manual; if your graphing utility has this capability, useit to graph each of the sequences in Exercises 89–92.What appears tobe happening to the terms of each sequence as gets larger?

89.

90.

91.

92.

Critical Thinking ExercisesMake Sense? In Exercises 93–96, determine whether eachstatement makes sense or does not make sense, and explainyour reasoning.

93. Now that I’ve studied sequences, I realize that the joke in thiscartoon is based on the fact that you can’t have a negativenumber of sheep.

an =

3n4+ n - 1

5n4+ 2n2

+ 1 n:30, 10, 14 by an :30, 1, 0.14

an =

2n2+ 5n - 7

n3 n:30, 10, 14 by an :30, 2, 0.24

an =

100n

n:30, 1000, 1004 by an :30, 1, 0.14

an =

n

n + 1 n:30, 10, 14 by an :30, 1, 0.14

n

e.a100,000 ,a10 , a100 , a1000 , a10,000 ,e L 2.7183e

an = a1 +

1nb

n

n

� SEQ �� SUM �

� SEQ �

54!154 - 32!3!

20!120 - 32!

94. By writing I can see that therange of a sequence is the set of positive integers.

95. It makes a difference whether or not I use parenthesesaround the expression following the summation symbol,

because the value of is 92, but the value of

is 43.

96. Without writing out the terms, I can see that in

causes the terms to alternate in sign.

In Exercises 97–100, determine whether each statement is true orfalse. If the statement is false, make the necessary change(s) toproduce a true statement.

97.

98. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, can be defined recursively using

where

99.

100.

101. Write the first five terms of the sequence whose first term is9 and whose general term is

for

Group Exercise102. Enough curiosities involving the Fibonacci sequence exist to

warrant a flourishing Fibonacci Association, which publishesa quarterly journal. Do some research on the Fibonaccisequence by consulting the Internet or the research depart-ment of your library, and find one property that interests you.After doing this research, get together with your group toshare these intriguing properties.

Preview ExercisesExercises 103–105 will help you prepare for the material coveredin the next section.

103. Consider the sequence Find and What do you observe?

104. Consider the sequence whose term is Find and What do youobserve?

105. Use the formula to find the eighthterm of the sequence 4, -3, -10, Á .

an = 4 + 1n - 121-72

a5 - a4 .a2 - a1 , a3 - a2 , a4 - a3 ,an = 4n - 3.nth

a5 - a4 . a3 - a2 , a4 - a3 ,a2 - a1 ,8, 3, -2, -7, -12, Á .

n Ú 2.

an = can - 1

2if an - 1 is even

3an - 1 + 5 if an - 1 is odd

a2

i = 1 aibi = a

2

i = 1 aia

2

i = 1 bi

a2

i = 1 1-12i2i

= 0

n Ú 2.an = an - 2 + an - 1 ,a0 = 1, a1 = 1;Á

n!1n - 12!

=

1n - 1

an =

1-122n

3n

1-122n

a8

i = 1 i + 7

a8

i = 1 1i + 72

a1 , a2 , a3 , a4 , Á , an , Á ,

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a-bltzmc11 967-1052-hr

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