sequences and series - novakmath.comnovakmath.com/9.1to9.3.pdf · ¥ use sequences and series to...

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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,

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,sequences are used to modelthe number of Best Buy storesfrom 1998 through 2003.

Sequences and Series

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 1Video

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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.

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

Video

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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.

nVideo

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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 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 1

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

�8 � 72 � 1

� 28

8!

2! � 6!�

8 � 7 � 6!2! � 6!

.

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 7Video

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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�

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 8Video

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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 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 2Number of

0 1 2 3blue cube faces

3 � 3 � 3

The symbol indicates an exercise in which you are instructed to use graphing technology or a symbolic computer algebra system.

Click on to view the complete solution of the exercise.

Click on to print an enlarged copy of the graph.

Click on to view the Make a Decision exercise.

Exercises

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 � 1�an ��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

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.n

ni�n

i�1 ai ,

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

na1, a2, a3, a4, . . .

Glossary

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

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

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

�1A � �2

128

15A � 3�1

57

B � 03

�1

410

0�2

2A � �1

50

41

�1

023,

B � 100

413

261A �

�241

�357

674,B � 0

8�12

11A � 10�4

76,

B � �26

4�3A � 6

453,

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

xi�2

.

�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

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 , anarithmetic sequence is used tomodel the seating capacity of an auditorium.

Arithmetic Sequences and Partial Sums

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

.

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 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 3. Substituting for in yields an alter-native 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

Video

viv00329a_l.html

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

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

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 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

Video

Simulation

si0088r1_l.html

viv00058a_l.html

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 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 4





Exercises

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, find thecommon 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, . . .


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

Glossary

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

12

18

24

30

n

an

2 106 84

2

4

6

8

10

n

an

84

2

4

6

8

10n

an

2 684

6

12

18

24

n

an

2 6

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 bowler fin-ishes.

(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

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 an arith-metic 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, click onthe Make a Decision button. (Data Source: NationalAssociation 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

Geometric SequencesIn the previous section, you learned that a sequence whose consecutive termshave a common difference is an arithmetic sequence. In this section, you willstudy another important type of sequence called a geometric sequence.Consecutive terms 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 be used to model and solve real-life problems. For instance,in Exercise 99, you will use a geometric sequence to modelthe population of China.

Geometric Sequences and Series

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

.

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 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 5

Example 2

Example 3

Video

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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

.

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

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 6Video

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.

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

Video

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.

.

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�

W RITING ABOUT MATHEMATICS

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 that the formula for compound interest is

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

.

Simulation

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Exercises

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

n

an

2 108

6

12

18

n

an

n2 4 6 108

750

600

450

300

150

an

2 106 84

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,. . .


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

Glossary

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 � . . .

9 � 6 � 4 �83 � . . .8 � 6 �

92 �

278 � . . .

��

n�0 �10�0.2�n�

�

n�0 �3�0.9�n

��

n�0 4�0.2�n�

�

n�0 �0.4�n

��

n�0 � 1

10�n��

n�0 4�1

4�n

��

n�0 2��2

3�n��

n�0 ��1

2�n

��

n�0 2�2

3�n��

n�0 �1

2�n

32 � 24 � 18 � . . . � 10.125

0.1 � 0.4 � 1.6 � . . . � 102.4

15 � 3 �35

� . . . �3

625

2 �12

�18

� . . . �1

2048

7 � 14 � 28 � . . . � 896

5 � 15 � 45 � . . . � 3645

�100

i�1 15�2

3�i�1�10

i�1 5��1

3�i�1

�25

i�0 8��1

2�i�10

i�1 8��1

4�i�1

�50

n�0

10�23�n�1�

40

n�0 2��1

4�n

�6

n�0 500�1.04�n�

5

n�0 300�1.06�n

�20

n�0

10�15�n�

15

n�0 2�4

3�n

�40

n�0 5�3

5�n�20

n�0 3�3

2�n

�12

i�1 16�1

2�i�1�6

i�1

32�14�i�1

�10

i�1 2�1

4�i�1�7

i�1 64��1

2�i�1

�8

n�1 5��3

2�n�1�9

n�1 ��2�n�1

�10

n�1 �5

2�n�1�9

n�1 2

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

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%

P � W�12

r ��1 � �1 �r

12��12t

.

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 � 4%, t � 40 yearsP � $75, r � 3%, t � 25 yearsP � $50, r � 7%, t � 20 years

tAr.

P

A �Per12�er t � 1�

er12 � 1.

A � Per12 � Pe 2r12 � . . . � Pe12tr12.tAr,

P

A � P��1 �r

12�12t

� 1�1 �12

r �.

P�1 �r

12�12t

.

A � P�1 �r

12� � P�1 �r

12�2

� . . . �

tAr,

P

A.

A � 50�1 �0.08

12 �1

� . . . � 50�1 �0.08

12 �60

.

A

A.

A � 100�1 �0.06

12 �1

� . . . � 100�1 �0.06

12 �60

.

A

Model It (cont inued)

(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.

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, click on the Makea Decision button. (Data Source: U.S. National ScienceFoundation)

8 �x � 1x � 4

�4

x � 1�

x � 4�x � 1��x � 4�

5 �7

x � 2�

2x � 2

x � 5x � 3

�10 � 2x2�3 � x�

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

�1 < r < 1.

�n � 1�

n

t � 1 � 2 ��

n �1

�0.9�n .

s2,

sn � 0 if t � �0.9�n�1sn � �16t 2 � 16�0.81�n�1,

.

.....

s4 � 0 if t � �0.9�3s4 � �16t 2 � 16�0.81�3,

s3 � 0 if t � �0.9�2s3 � �16t 2 � 16�0.81�2,

s2 � 0 if t � 0.9s2 � �16t 2 � 16�0.81�,s1 � 0 if t � 1s1 � �16t 2 � 16,

0.81hh

n � 4n

. . . ,n �an � 54.6e0.172n,

an

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