Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

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770 CHAPTER 9 Additional Topics in Algebra 9–10

College Algebra Graphs & Models—

2. A series is the of the numbers from a

given sequence.

䊳

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

9.1 EXERCISES

1. A sequence is a(n) of numbers listed in a

speciﬁc .

4. The notation n! represents the of the

natural number n, with all those n.

3. A sequence that uses the preceding term(s) to generate

those that follow is called a sequence.

6. Describe the characteristics of an alternating

sequence and give one example.

5. Describe the characteristics of a recursive sequence

and give one example.

䊳

DEVELOPING YOUR SKILLS

Find the ﬁrst four terms, then ﬁnd the 8th and 12th

term for each nth term given.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

Use a calculator to: (a) ﬁnd the indicated term for each

sequence, and (b) generate the ﬁrst ﬁve terms of each

sequence and store the results in a list. Use fractions if

possible; round to tenths when necessary.

23. 24.

25. 26.

27. 28. a

⫽ 3a

n⫺1

; a

⫽ 2a

n⫺1

; a

⫽

1⫺12

n⫹1

2n ⫺ 1

; a

⫽

1⫺12

n⫹1

; a

⫽ 1n ⫺ 22

; a

⫽ n

⫺ 2; a

⫽ 1⫺12

⫺n

⫽ 1⫺12

⫽

1⫺12

n⫹1

⫺ 1

⫽

1⫺12

n1n ⫹ 12

⫽

⫽ a

⫽ a1 ⫹

⫽

n ⫹ 1

⫽

1⫺12

⫽ 1⫺12

⫽ 2n

⫺ 12a

⫽ 3n

⫺ 3

⫽ 2n ⫹ 3a

⫽ 2n ⫺ 1

29. 30.

31.

32.

Find the ﬁrst ﬁve terms of each recursive sequence.

33. 34.

35. 36.

37. 38.

Simplify each factorial expression.

39. 40. 41.

42. 43. 44.

Write out the ﬁrst four terms in each sequence.

45. 46. a

⫽

1n ⫹ 32!

⫽

1n ⫹ 12!

10!

3!7!

2!6!

3!3!

7!2!

12!

10!

⫽ 1, c

⫽ 2

⫽ c

n⫺1

⫹ 1c

n⫺2

•

⫽ 64, c

⫽ 32

⫽

n⫺2

⫺ c

n⫺1

⫽⫺2

⫽ a

n⫺1

⫺ 16

⫽⫺1

⫽ 1a

n⫺1

⫹ 3

⫽ 3

⫽ 2a

n⫺1

⫺ 3

⫽ 2

⫽ 5a

n⫺1

⫺ 3

⫽

12n ⫺ 1212n ⫹ 12

; a

⫽

n12n ⫹ 12

; a

⫽ an ⫹

; a

⫽ a1 ⫹

; a

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9–11 Section 9.1 Sequences and Series 771

College Algebra Graphs & Models—

47. 48.

49. 50.

Find the indicated partial sum for each sequence.

51. 52.

53. 54.

55. 56.

Expand and evaluate each series. Verify results using a

graphing calculator.

57. 58. 59.

60. 61. 62.

63. 64. 65.

66. 67. 68.

Write each sum using sigma notation. Answers are not

necessarily unique.

69. a.

b. 5 ⫹ 10 ⫹ 15 ⫹ 20 ⫹ 25

4 ⫹ 8 ⫹ 12 ⫹ 16 ⫹ 20

兺

k⫽2

1⫺12

k⫹1

⫺ 1

兺

k⫽3

1⫺12

k1k ⫺ 22

兺

j⫽3

兺

j⫽3

兺

i⫽2

兺

i⫽1

兺

k⫽1

1⫺12

兺

k⫽1

1⫺12

兺

k⫽1

⫹ 12

兺

k⫽1

12k

⫺ 32

兺

i⫽1

12i ⫺ 32

兺

i⫽1

13i ⫺ 52

⫽

n ⫹ 1

; S

⫽

; S

⫽ 3n ⫺ 1; S

⫽ 2n ⫺ 1; S

⫽ n

; S

⫽ n; S

⫽

1n ⫹ 32!

12n2!

⫽

1n ⫹ 12!

13n2!

70. a.

71. a.

72. a.

For the given general term a

, write the indicated sum

using sigma notation.

73.

74.

75. third partial sum

76. sixth partial sum

77. sum for to 7

78. sum for to 6

Compute each sum by applying properties of summation.

79. 80.

81. 82.

兺

k⫽1

12k

⫹ 52

兺

k⫽1

13k

⫹ k2

兺

i⫽1

13 ⫹ 2i2

兺

i⫽1

14i ⫺ 52

n ⫽ 2a

⫽ n

;

n ⫽ 3a

⫽

;

⫽ 2n ⫺ 1;

⫽

;

⫽

⫹ 1

n ⫹ 1

; S

⫽ n ⫹ 3; S

1 ⫹

⫹

120

⫹

720

⫹

0.1 ⫹ 0.01 ⫹ 0.001 ⫹ 0.0001 ⫹

1 ⫹

⫹

1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫹ 11 ⫹

1 ⫺ 8 ⫹ 27 ⫺ 64 ⫹ 125 ⫺ 216

⫺1 ⫹ 4 ⫺ 9 ⫹ 16 ⫺ 25 ⫹ 36

83. Sum of

The sum of the ﬁrst n terms of the sequence

deﬁned by 4, 7, 10, ,

is given by the formula shown. Find

using the formula, then verify by direct

calculation.

p13n ⫺ 22,

⫽ 3n ⫺ 2 ⫽ 1,

ⴝ 3n ⴚ 2: S

ⴝ

n13n ⴚ 12

84. Sum of

The sum of the ﬁrst n terms of the sequence deﬁned

by 5, 8, 11, , is

given by the formula shown. Find S

using the

formula, then verify by direct calculation. Observing

the formulas from Exercises 83 and 84, can you

now state the sum formula for a

⫽ 3n ⫺ 0?

p13n ⫺ 12,pa

⫽ 3n ⫺ 1 ⫽ 2,

ⴝ 3n ⴚ 1: S

ⴝ

n13n ⴙ 12

䊳

WORKING WITH FORMULAS

85. Wage increases: Latisha gets $7.25 an hour for ﬁlling candy machines for Archtown Vending. Each year

she receives a $0.50 hourly raise. List Latisha’s hourly wage for the ﬁrst 5 yr. How much will she make in the

ﬁfth year if she works 8 hr per day for 240 working days?

86. Average birth weight: The average birth weight of a certain animal species is 900 g, with the baby gaining

125 g each day for the ﬁrst 10 days. List the infant’s weight for the ﬁrst 10 days. How much does the infant

weigh on the 10th day?

䊳

APPLICATIONS

Use the information given in each exercise to determine the nth term a

for the sequence described. Then use the nth

term to list the speciﬁed number of terms.

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772 CHAPTER 9 Additional Topics in Algebra 9–12

College Algebra Graphs & Models—

87. Blue-book value: Steve’s car has a blue-book

value of $6000. Each year it loses 20% of its value

(its value each year is 80% of the year before). List

the value of Steve’s car for the next 5 yr.

(Hint: .)

88. Effects of inﬂation: Suppose inﬂation (an increase

in value) will average 4% for the next 5 yr. List the

growing cost (year by year) of a DVD that costs

$15 right now. (Hint: )

89. Stocking a lake: A local ﬁshery stocks a large

lake with 1500 bass and then adds an additional

100 mature bass per month until the lake nears

maximum capacity. If the bass population grows

at a rate of 5% per month through natural

reproduction, the number of bass in the pond

after n months is given by the recursive sequence

How many

bass will be in the lake after 6 months?

⫽ 1500, b

⫽ 1.05b

n⫺1

⫹ 100.

⫽ 15.

⫽ 6000

90. Species preservation: The Interior Department

introduces 50 wolves (male and female) into a

large wildlife area in an effort to preserve the

species. Each year about 12 additional adult

wolves are added from capture and relocation

programs. If the wolf population grows at a rate

of 10% per year through natural reproduction,

the number of wolves in the area after n years

is given by the recursive sequence

How many wolves

are in the wildlife area after 6 years?

Use your calculator to ﬁnd the partial sums for

and for the summations given,

and attempt to name the number the summation

approximates:

91. 92.

兺

k⫽1

⫹ 3

兺

k⫽1

n ⴝ 12n ⴝ 4, n ⴝ 8,

⫽ 50, w

⫽ 1.10w

n⫺1

⫹ 12.

䊳

EXTENDING THE CONCEPT

93. Verify that a constant can be factored out of a sum.

That is, verify that the following statement is true:

兺

j⫽1

⫽ c

兺

j⫽1

94. Verify that a summation may be distributed to two

(or more) sequences. That is, verify that the

following statement is true:

兺

i⫽1

⫾ b

2⫽

兺

i⫽1

⫾

兺

i⫽1

䊳

MAINTAINING YOUR SKILLS

97. (5.3) Write in exponential form, then

solve by equating bases.

98. (3.6) Set up the difference quotient for

then rationalize the numerator.

1x2⫽ 1x,

log

⫽⫺x

99. (8.4) Solve the nonlinear system.

100. (7.3) Solve the system using a matrix equation.

•

25x ⫹ y ⫺ 2z ⫽⫺14

2x ⫺ y ⫹ z ⫽ 40

⫺7x ⫹ 3y ⫺ z ⫽⫺13

⫹ y

⫽ 9

⫺ 4x

⫽ 16

95. a

⫽

n1n ⫹ 121n ⫹ 22

96. a

⫽

12n ⫺ 1212n ⫹ 12

Regarding Exercises 91 and 92, sometimes a series will approach a ﬁxed number very slowly, and many more terms

must be added before this value is recognized. Use your graphing calculator to compute the sums S

, S

, and S

for

the following sequences to see if you can recognize the number. Add more terms if necessary.

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College Algebra Graphs & Models—

Similar to the way polynomials fall into certain groups or families (linear, quadratic,

cubic, etc.), sequences and series with common characteristics are likewise grouped. In

this section, we focus on sequences where each successive term is generated by adding

a constant value, as in the sequence 1, 8, 15, 22, 29, , where 7 is added to a given

term in order to produce the next term.

A. Identifying an Arithmetic Sequence and Finding

the Common Difference

An arithmetic sequence is one where each successive term is found by adding a ﬁxed

constant to the preceding term. For instance 3, 7, 11, 15, is an arithmetic sequence,

since adding 4 to any given term produces the next term. This also means if you take

the difference of any two consecutive terms, the result will be 4 and in fact, 4 is called

the common difference d for this sequence. Using the notation developed earlier, we

can write where a

represents any term of the sequence and rep-

resents the term that follows a

Arithmetic Sequences

Given a sequence a

, a

, , a

, where k, and

if there exists a common difference d such that for all k,

then the sequence is an arithmetic sequence.

The difference of successive terms can be rewritten as (for

to highlight that each following term is found by adding d to the previous term.

EXAMPLE 1

䊳

Identifying an Arithmetic Sequence

Determine if the given sequence is arithmetic. If yes, name the common difference.

If not, try to determine the pattern that forms the sequence.

a. 2, 5, 8, 11, b.

Solution

䊳

a. Begin by looking for a common difference Checking each pair

of consecutive terms we have

This is an arithmetic sequence with common difference

b. Checking each pair of consecutive terms yields

Since the difference is not constant, this is not an arithmetic sequence. It

appears the sequence is formed by adding to each previous term, for natural

numbers k.

Now try Exercises 7 through 18

䊳

⫽

⫺

⫽

⫺

⫽

⫺

⫽

⫺

d ⫽ 3.

5 ⫺ 2 ⫽ 3

8 ⫺ 5 ⫽ 3

11 ⫺ 8 ⫽ 3

and so on.

d ⫽ a

k⫹1

⫺ a

, pp

k ⱖ 12a

k⫹1

⫽ a

⫹ d

k⫹1

⫺ a

⫽ d

k 6 n,n 僆⺞pa

k⫹1

d ⫽ a

k⫹1

⫺ a

LEARNING OBJECTIVES

In Section 9.2 you will see

how we can:

A. Identify an arithmetic

sequence and its common

difference

B. Find the n th term of an

arithmetic sequence

C. Find the n th partial sum

of an arithmetic sequence

D. Solve applications

involving arithmetic

sequences

9–13 773

9.2 Arithmetic Sequences

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774 CHAPTER 9 Additional Topics in Algebra 9–14

College Algebra Graphs & Models—

EXAMPLE 2

䊳

Writing the First

Terms of an Arithmetic Sequence

Write the ﬁrst ﬁve terms of the arithmetic sequence, given the ﬁrst term a

and the

common difference d.

a. and b. and

Solution

䊳

a. and Starting at add to each new term to

generate the sequence: 12, 8, 4, 0, .

b. and Starting at and adding to each new term will

generate the sequence: . Note that since the common denominator is

6, terms of the sequence can quickly be found by adding to the previous

term and reducing if possible.

Now try Exercises 19 through 30

䊳

B. Finding the

th Term of an Arithmetic Sequence

If the values a

and d from an arithmetic sequence are known, we could generate the

terms of the sequence by adding multiples of d to the ﬁrst term, instead of adding d to

each new term. For example, we can generate the sequence 3, 8, 13, 18, 23 by adding

multiples of 5 to the ﬁrst term

current term initial coefﬁcient of common

term difference

It’s helpful to note the coefﬁcient of d is 1 less than the subscript of the current

term (as shown): This observation leads us to a formula for the nth term.

The

th Term of an Arithmetic Sequence

The nth term of an arithmetic sequence is given by

where d is the common difference.

EXAMPLE 3

䊳

Finding a Speciﬁed Term in an Arithmetic Sequence

Find the 24th term of the sequence 0.1, 0.4, 0.7, 1, .

Solution

䊳

Instead of creating all terms up to the 24th, we determine the constant d and use

the nth term formula. By inspection we note and

th term formula

substitute 0.1 for

and 0.3 for

eliminate parentheses

simplify

To ﬁnd the 24th term we substitute 24 for n:

substitute 24 for

result

Now try Exercises 31 through 42

䊳

⫽ 7.0

⫽ 0.31242⫺ 0.2

⫽ 0.3n ⫺ 0.2

⫽ 0.1 ⫹ 0.3n ⫺ 0.3

⫽ 0.1 ⫹ 1n ⫺ 120.3

⫽ a

⫹ 1n ⫺ 12d

d ⫽ 0.3.a

⫽ 0.1

⫽ a

⫹ 1n ⫺ 12d

5 ⫺ 1 ⫽ 4.

23 ⫽ 3 ⫹ 1425

⫽ a

⫹ 4d

18 ⫽ 3 ⫹ 1325

⫽ a

⫹ 3d

13 ⫽ 3 ⫹ 1225

⫽ a

⫹ 2d

8 ⫽ 3 ⫹ 1125

⫽ a

⫹ 1d

3 ⫽ 3 ⫹ 1025

⫽ a

⫹ 0d

⫽ 3:

⫽

d ⫽

⫽

⫺4

⫺4a

⫽ 12,d ⫽⫺4.a

⫽ 12

d ⫽

⫽

d ⫽⫺4a

⫽ 12

A. You’ve just seen how

we can identify an arithmetic

sequence and its common

difference

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9–15 Section 9.2 Arithmetic Sequences 775

College Algebra Graphs & Models—

EXAMPLE 4

䊳

Determining the Number of Terms in an Arithmetic Sequence

Find the number of terms in the arithmetic sequence 2, ,

Solution

䊳

By inspection we see that and As before,

th term formula

substitute 2 for

and ⫺7 for

distribute ⫺7

simplify

Although we don’t know the number of terms in the sequence, we do know the last

or nth term is Substituting for a

gives

substitute ⫺411 for

solve for

There are 60 terms in this sequence.

Now try Exercises 43 through 50

䊳

Note that in both Examples 3 and 4, the nth term had the form of a linear equation

after simplifying: and . This is a char-

acteristic of arithmetic sequences, with the common difference d corresponding to

the slope m. This means the graph of an arithmetic sequence will always be a set of

discrete points that lie on a straight line. After entering (from

Example 3), Figure 9.23 shows the table of values for , with the graph

in Figure 9.24A.

In sequence , we still set the size of the viewing

window as before, but we can also stipulate the range

of values of n to be used (nMin and nMax), which

term we want to plot as a beginning (PlotStart), and

whether we want all following terms graphed (Plot-

Step ), every second term graphed (PlotStep ),

and so on. The graph in Figure 9.24A was gener-

ated using the values shown in Figure 9.24B

( cannot be seen).

To see the graph of a sequence more distinctly, we can enter the natural numbers 1

through 10 in L1, then deﬁne L2 as u(L1) (Figures 9.25 and 9.26) and plot these points

(Figure 9.27).

Ymin ⫽⫺1.5, Ymax ⫽ 5, and Yscl ⫽ 1

⫽ 2⫽ 1

MODE

Figure 9.23

⫺1.5

Figure 9.24A

⫽ 0.3n ⫺ 0.2

u1n2⫽ 0.3n ⫺ 0.2

⫽⫺7n ⫹ 9a

⫽ 0.3n ⫺ 0.21y ⫽ mx ⫹ b2

60 ⫽ n

⫺411 ⫽⫺7n ⫹ 9

⫺411⫺411.

⫽⫺7n ⫹ 9

⫽ 2 ⫺ 7n ⫹ 7

⫽ 2 ⫹ 1n ⫺ 121⫺72

⫽ a

⫹ 1n ⫺ 12d

d ⫽⫺7.a

⫽ 2

⫺411.p⫺19,⫺12,⫺5,

Figure 9.24B

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776 CHAPTER 9 Additional Topics in Algebra 9–16

College Algebra Graphs & Models—

Figure 9.29Figure 9.28

One additional advantage of this approach is that we can easily verify the common dif-

ference is d using the “ List(” feature, which automatically computes the difference

between each successive term in a speciﬁed list. This option is located in the same sub-

menu as the “seq(” option, accessed using (LIST) (OPS) 7: List(L2)

. See Figures 9.28 and 9.29.

ENTER

STAT2nd

EXAMPLE 5

䊳

Graphing Arithmetic Sequences

Enter the natural numbers 1 through 6 in L1, and the terms of the sequence ⫺0.45,

⫺0.1, 0.25, 0.6, 0.95, 1.3 in L2. Then verify the sequence in L2 is arithmetic by

a. Graphing the related points to see if they appear linear.

b. Using the List( feature.

c. Finding the nth term that deﬁnes the sequence and graph the sequence.

Solution

䊳

a. The plotted points are shown in Figure 9.30 and appear to be linear.

b. The List( feature shows there is a common difference of 0.35 (Figure 9.31).

c. With and , the nth term must be

th term formula

substitute for

and

simplify

The nth term for this sequence is . The graph is shown in

Figure 9.32.

Now try Exercises 51 through 54

䊳

⫽ 0.35n ⫺ 0.8

⫽⫺0.45 ⫹ 1n ⫺ 1210.352

⫽ a

⫹ 1n ⫺ 12d

d ⫽ 0.35a

⫽⫺0.45

⫺1

Figure 9.32

Figure 9.31

⫺1

Figure 9.30

Figure 9.26

⫺1.5

Figure 9.27

Figure 9.25

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9–17 Section 9.2 Arithmetic Sequences 777

CCoolllleeggee AAllggeebbrraa GGrraapphhss && MMooddeellss——

If the term a

is unknown but a term a

is given, the nth term can be written

(the subscript of the term a

and coefﬁcient of d sum to n).

EXAMPLE 6

䊳

Finding the First Term of an Arithmetic Sequence

Given an arithmetic sequence where and ﬁnd the common

difference d and the value of a

Solution

䊳

At ﬁrst it seems that not enough information is given, but recall we can express a

as the sum of any earlier term and the appropriate multiple of d. Since a

is known,

we write (note as required).

is unknown

substitute 0.9 for

and 0.55 for

subtract 0.55

solve for

Having found d, we can now solve for a

th term formula for

substitute 0.9 for

and 0.05 for

simplify

solve for

The ﬁrst term is and the common difference is

Now try Exercises 55 through 60

䊳

d ⫽ 0.05.a

⫽ 0.3

0.9 ⫽ a

⫹ 0.6

0.9 ⫽ a

⫹ 1210.052

n ⫽ 13 a

⫽ a

⫹ 12d

d ⫽ 0.05

0.35 ⫽ 7d

0.9 ⫽ 0.55 ⫹ 7d

⫽ a

⫹ 7d

13 ⫽ 6 ⫹ 7a

⫽ a

⫹ 7d

⫽ 0.9,a

⫽ 0.55

⫽ a

⫹ 1n ⫺ k2d

B. You’ve just seen how

we can ﬁnd the nth term of an

arithmetic sequence

C. Finding the nth Partial Sum of an Arithmetic Sequence

Using sequences and series to solve applications often requires computing the sum of

a given number of terms. To develop and understand the approach used, consider the

sum of the ﬁrst 10 natural numbers. Using S

to represent this sum, we have

. We could just use brute force,

but if we rewrite the sum a second time but in reverse order, then add it to the ﬁrst, we

ﬁnd that each column adds to 11.

⫽ 1 ⫹ 2 ⫹ 3 ⫹ 4 ⫹ 5 ⫹ 6 ⫹ 7 ⫹ 8 ⫹ 9 ⫹ 10

⫽ 10 ⫹ 9 ⫹ 8 ⫹ 7 ⫹ 6 ⫹ 5 ⫹ 4 ⫹ 3 ⫹ 2 ⫹ 1

⫽ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11 ⫹ 11

Since there are 10 columns, the total is 110 but this is twice the actual sum

and we ﬁnd that .

Now consider the sequence a

, a

, , a

with common difference d. Use S

to represent the sum of the ﬁrst n terms and write the original series, then the series in

reverse order underneath. Since one row increases at the same rate the other decreases,

the sum of each column remains constant, and for simplicity’s sake we choose

to represent this sum.

To understand why each column adds to , consider the sum in the second

column: . From and , we obtain

by adding the equations, which gives a result of . Since

there are n columns, we end up with and solving for S

gives the

formula for the ﬁrst n terms of an arithmetic sequence.

⫽ n1a

⫹ a

⫹ d2⫹ 1a

⫺ d2

⫹ a

n⫺1

⫽a

n⫺1

⫽ a

⫺ da

⫽ a

⫹ da

⫹ a

n⫺1

⫹ a

⫽ a

⫹ a

⫹

⫹ a

n⫺2

⫹ a

n⫺1

⫹ a

⫽ a

⫹ a

n⫺1

⫹ a

n⫺2

⫹

⫹ a

⫽ 1a

⫹ a

2⫹ 1a

⫹ a

2⫹ 1a

⫹ a

2⫹

⫹ 1a

⫹ a

2⫹ 1a

⫹ a

2⫹ 1a

⫹ a

⫽ 110, so S

⫽ 55

11 ⫻ 10 ⫽

⫽ 1 ⫹ 2 ⫹ 3 ⫹ 4 ⫹ 5 ⫹ 6 ⫹ 7 ⫹ 8 ⫹ 9 ⫹ 10

add

columns

vertically

↓

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The nth Partial Sum of an Arithmetic Sequence

Given an arithmetic sequence with ﬁrst term a

, the nth partial sum is given by

In words: The sum of an arithmetic sequence is one-half the number of terms times the

sum of the ﬁrst and last term.

EXAMPLE 7

䊳

Computing the Sum of an Arithmetic Sequence

Use the summation formula to ﬁnd the sum of the ﬁrst 75 positive odd integers:

. Verify the result using a graphing calculator.

Solution

䊳

The initial terms of the sequence are 1, 3, 5, and we note and

To use the sum formula, we need the value of :

formula shows so

sum formula

substitute 75 for

substitute 1 for

, 149 for

result

The sum of the ﬁrst 75 positive odd integers is 5625.

To verify, we enter on the

screen, and ﬁnd the sum of the ﬁrst 75 terms of the

sequence on the home screen as before. See ﬁgure.

Now try Exercises 61 through 66

䊳

By substituting the nth term formula directly into the formula for partial sums,

we’re able to ﬁnd a partial sum without actually having to ﬁnd the nth term:

sum formula

substitute for

alternative formula for the

th partial sum

See Exercises 67 through 72 for more on this alternative formula.

⫽

32a

⫹ 1n ⫺ 12d4

⫹ 1n ⫺ 12d ⫽

⫹ 3a

⫹ 1n ⫺ 12d42

⫽

⫹ a

u1n2⫽ 2n ⫺ 1

⫽ 5625

⫽

11 ⫹ 1492

⫽

⫹ a

⫽

⫹ a

⫽ 149.a

⫽ a

⫹ 74d ⫽ 1 ⫹ 74122,

21752⫺ 1 ⫽ 149.a

n ⫽ 75.

⫽ 1, d ⫽ 2,p

兺

n⫽1

12n ⫺ 12

⫽

⫹ a

778 CHAPTER 9 Additional Topics in Algebra 9–18

College Algebra Graphs & Models—

C. You’ve just seen how

we can ﬁnd the nth partial sum

of an arithmetic sequence

cob19545_ch09_774-782.qxd 12/17/10 9:36 PM Page 778

D. Applications

In the evolution of certain plants and shelled animals,

sequences and series seem to have been one of nature’s

favorite tools. The sprials found on many ferns and other

plants are excellent examples of sequences in nature, as are

the size of the chambers in nautilus shells (see Figures 9.33

and 9.34). Sequences and series also provide a good mathe-

matical model for a variety of other situations as well.

EXAMPLE 8

䊳

Solving an Application of Arithmetic Sequences: Seating Capacity

Cox Auditorium is an amphitheater that has 40 seats in the ﬁrst row, 42 seats in the

second row, 44 in the third, and so on. If there are 75 rows in the auditorium, what

is the auditorium’s seating capacity?

Solution

䊳

The number of seats in each row gives the terms of an arithmetic sequence with

and To ﬁnd the seating capacity, we need to ﬁnd the total

number of seats, which is the sum of this arithmetic sequence. Since the value of

is unknown, we opt for the alternative formula

sum formula

substitute 40 for

, 2 for

, and 75 for

simplify

result

The seating capacity of Cox Auditorium is 8550.

Now try Exercises 75 through 80

䊳

⫽ 8550

⫽

12282

⫽

321402⫹ 175 ⫺ 121224

⫽

32a

⫹ 1n ⫺ 12d4

⫽

32a

⫹ 1n ⫺ 12d4.

n ⫽ 75.d ⫽ 2,a

⫽ 40,

9–19 Section 9.2 Arithmetic Sequences 779

College Algebra Graphs & Models—

D. You’ve just seen how

we can solve applications

involving arithmetic sequences

Figure 9.34

Figure 9.33

spiral fern

nautilus

2. The sum of the ﬁrst n terms of an arithmetic

sequence is called the nth .

䊳

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

1. Consecutive terms in an arithmetic sequence differ

by a constant called the .

3. The formula for the nth partial sum of an

arithmetic sequence is , where a

is the term.

⫽

4. The nth term formula for an arithmetic sequence is

, where a

is the term

and d is the .

⫽

5. Discuss how the terms of an arithmetic sequence

can be written in various ways using the

relationship a

⫽ a

⫹ 1n ⫺ k2d.

6. Describe how the formula for the nth partial sum

was derived, and illustrate its application using a

sequence from the exercise set.

9.2 EXERCISES

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