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

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Solution

䊳

a. The lengths of each swing form the terms of a

geometric sequence with and .

The ﬁrst few terms are 2, 1.8, 1.62, 1.458, and

so on. For the 8th term we have:

th term formula

substitute 8 for

, 2 for

, and 0.9 for

The pendulum travels about 0.957 m on its

8th swing. See Figure 9.48.

b. For the total distance traveled after eight

swings, we compute the value of S

th partial sum formula

The pendulum has traveled about 11.4 m by

the end of the 8th swing. See Figure 9.49.

c. To ﬁnd the number of swings until the length

of each swing is less than 0.5 m, we solve for

n in the equation . This yields

divide by 2

After the 14th swing, each successive swing will be less than 0.5 m. See

Figure 9.50.

d. For the total distance traveled before coming

to rest, we consider the related inﬁnite

geometric series, with and

inﬁnite sum formula

substitute 2 for

and 0.9 for

result

The pendulum would travel 20 m before coming to rest. Note that summing a

larger number of terms on a calculator takes an increasing amount of time. The

values for S

and S

150

are shown in Figure 9.51.

Now try Exercises 111 and 112

䊳

As mentioned in Section 9.1, sometimes the sequence or series for a particular

application will use the preliminary or inaugural term a

, as when an initial amount of

money is deposited before any interest is earned, or the efﬁciency of a new machine

after purchase—prior to any wear and tear.

⫽ 20

⫽

1 ⫺ 0.9

⫽

1 ⫺ r

r ⫽ 0.9.a

⫽ 2

14.16 ⬇ n

ln 0.25

ln 0.9

⫹ 1 ⫽ n

ln 0.25 ⫽ 1n ⫺ 12ln 0.9

0.25 ⫽ 10.92

n⫺1

0.5 ⫽ 210.92

n⫺1

⬇ 11.4

⫽

211 ⫺ 0.9

1 ⫺ 0.9

⫽

11 ⫺ r

1 ⫺ r

⬇ 0.957

⫽ 210.92

8⫺1

⫽ a

n⫺1

r ⫽ 0.9a

⫽ 2

790 CHAPTER 9 Additional Topics in Algebra 9–30

College Algebra Graphs & Models—

Figure 9.48

Figure 9.49

Figure 9.50

Figure 9.51

substitute 2 for

0.9 for

and 8 for

take the natural log,

apply power property

solve for

(exact form)

solve for

(approximate form)

cob19545_ch09_783-796.qxd 11/9/10 9:11 PM Page 790

EXAMPLE 10

䊳

Equipment Efﬁciency—Furniture Manufacturing

The manufacturing of mass-produced furniture requires robotic machines to drill

numerous holes for the bolts used in the assembly process. When new, the drill bits

are capable of drilling through hardwood at a rate of 6 cm/sec. As the bit becomes

worn, it loses 4% of its drilling speed per day.

a. How many cm/sec can the bit drill through after a 5-day workweek?

b. When the drilling speed falls below 3.6 cm/sec, the bit must be replaced.

After how many days must the bit be replaced?

Solution

䊳

The efﬁciency of a new drill bit (prior to use) is given as . Since

the bit loses of its efﬁciency per day, it maintains of its

efﬁciency, showing that after 1 day of use . This means the nth

term formula will be .

a. At the end of day 5 we have

After 5 days, the bit can drill through the hardwood at about 4.9 cm/sec.

b. To ﬁnd the number of days until the efﬁciency falls below 3.6 cm/sec, we

replace a

with 3.6 and solve for n.

substitute 3.6 for

divide

take the natural log of both sides

power property

divide

solve for

(exact form)

solution (approximate form)

The drill bit must be replaced after 12 full days of use.

Now try Exercises 113 through 126

䊳

12.5 ⬇ n

ln 0.625

ln 0.96

⫹ 1 ⫽ n

ln 0.625

ln 0.96

⫽ n ⫺ 1

ln 0.625 ⫽ 1n ⫺ 12

ln 0.96

ln 0.625 ⫽ ln 0.96

n⫺1

0.625 ⫽ 0.96

n⫺1

3.6 ⫽ 5.7610.962

n⫺1

⬇ 4.9

⫽ 5.7610.962

5⫺1

⫽ 5.7610.962

n⫺1

⫽ 0.96162⫽ 5.76

96% ⫽ 0.964% ⫽ 0.04

⫽ 6 cm/sec

9–31 Section 9.3 Geometric Sequences 791

College Algebra Graphs & Models—

E. You’ve just seen how

we can solve application

problems involving geometric

sequences and series

4. For the general sequence a

, a

, , the

ﬁfth partial sum is given by .

5. Describe/Discuss how the formula for the nth

partial sum is related to the formula for the sum of

an inﬁnite geometric series.

6. Describe the difference(s) between an arithmetic

and a geometric sequence. How can a student

prevent confusion between the formulas?

⫽

, p ,p

䊳

CONCEPTS AND VOCABULARY

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

9.3 EXERCISES

1. In a geometric sequence, each successive term is

found by the preceding term by a ﬁxed

value r.

2. In a geometric sequence, the common ratio r can

be found by computing the of any two

consecutive terms.

3. The nth term of a geometric sequence is given by

, for any n ⱖ 1.

⫽

cob19545_ch09_783-796.qxd 12/17/10 12:50 AM Page 791

792 CHAPTER 9 Additional Topics in Algebra 9–32

College Algebra Graphs & Models—

䊳

DEVELOPING YOUR SKILLS

Determine if the sequence given is geometric. If yes,

name the common ratio. If not, try to determine the

pattern that forms the sequence.

7. 4, 8, 16, 32,

8. 2, 6, 18, 54, 162,

9. 3, 12, 48,

10. 128, 8,

11. 2, 5, 10, 17, 26,

12. 3,

13. 3, 0.3, 0.03, 0.003,

. 12, 0.12, 0.0012, 0.000012,

15. 3, 60,

16. 2, 40,

17. 25, 10, 4,

18. 24,

19.

20.

21.

22.

23. 240, 120, 40, 10, 2,

24.

Write the ﬁrst four terms of the sequence, given a

and r.

25. 26.

27. 28.

29. 30.

31. 32.

Write the expression for the nth term, then ﬁnd the

indicated term for each sequence.

33. ﬁnd a

34. ﬁnd a

35. ﬁnd a

36. ﬁnd a

37. ﬁnd a

38. ﬁnd a

⫽ 13, r ⫽ 13;

⫽ 2, r ⫽ 12;

⫽

, r ⫽ 4;

⫽⫺

, r ⫽⫺5;

⫽ 48, r ⫽⫺

;

⫽⫺24, r ⫽

;

⫽ 0.024, r ⫽ 0.01a

⫽ 0.1, r ⫽ 0.1

⫽ 15, r ⫽ 15a

⫽ 4, r ⫽ 13

⫽

, r ⫽

⫽⫺6, r ⫽⫺

⫽ 2, r ⫽⫺4a

⫽ 5, r ⫽ 2

⫺120, ⫺60, ⫺20, ⫺5, ⫺1, p

, p

192

, p

⫺16,

, p⫺36,

, p

p⫺240,⫺8,⫺

p⫺360,⫺12,⫺1,

p⫺13, ⫺9, ⫺5, ⫺1,

p⫺2,⫺32,

p⫺24,⫺6,

Identify a

and r, then write the expression for the nth

term and use it to ﬁnd a

, a

, and a

39. 40.

41. 729, 243, 81, 27, 9, 42. 625, 125, 25, 5, 1,

43.

44. 36, 12,

45. 0.2, 0.08, 0.032, 0.0128,

46. 0.5, 0.245,

Find the number of terms in each sequence.

47.

48.

49.

50.

51.

52.

53. 2, 18,

54. 3, 12,

55. 64, 32,

56. 243, 81,

57.

58.

For Exercises 59 through 62, enter the natural numbers

1 through 6 in L1 on a graphing calculator, and the

terms of the given sequence in L2. Then determine if the

sequence is geometric by (a) graphing the related points

to see if they appear to lie on an exponential curve, and

(b) computing the successive ratios of all terms. If a

geometric sequence, ﬁnd the nth term and graph the

sequence.

59. 131.25, 26.25, 5.25, 1.05, 0.21, 0.042,

60. 2, , 10, , 50, ,

61. 20, 16, 12, 8, 4, 0,

62. , 1,

Find the common ratio r and the value of a

using the

information given (assume ).

63. 64.

65. 66.

67. 68. a

⫽

, a

⫽ 25a

⫽

, a

⫽ 54

⫽

, a

⫽

, a

⫽

⫽ 6, a

⫽ 486a

⫽ 324, a

⫽ 64

r 7 0

p5015

1025225

⫺

, ⫺

, ⫺5, p , ⫺135

, ⫺

, ⫺3, p , 96

2713

, p , 18113,

1612

, p , 13212,

⫺24, p , ⫺6144⫺6,

⫺54, p , ⫺4374⫺6,

⫽ 2, a

⫽ 1458, r ⫽⫺13

⫽⫺1, a

⫽⫺1296, r ⫽ 16

⫽ 4, a

⫽

512

, r ⫽

⫽ 16, a

⫽

, r ⫽

⫽ 1, a

⫽⫺128, r ⫽⫺2

⫽ 9, a

⫽ 729, r ⫽ 3

⫺0.1715, p⫺0.35,

413

, p1213,3613,

, 1, 12, 2, p

⫺

, ⫺

, 7, ⫺14, p

, ⫺

, ⫺1, 3, p

ⴝ a

nⴚ1

cob19545_ch09_783-796.qxd 11/9/10 9:11 PM Page 792

Find the partial sum indicated.

69. ﬁnd S

70. ﬁnd S

71. ﬁnd S

72. ﬁnd S

73. ﬁnd S

74. ﬁnd S

75. ; ﬁnd S

76. ﬁnd S

77. ﬁnd S

78. ﬁnd S

79. ﬁnd S

80. ﬁnd S

Find the partial sum indicated, and verify the result

using a graphing calculator. For Exercises 85 and 86,

use the summation properties from Section 9.1.

81. 82.

83. 84.

85. 86.

Find the indicated partial sum using the information

given. Write all results in simplest form.

87. ﬁnd S

88. ﬁnd S

⫽ 1, a

⫽⫺27;

⫽⫺5, a

⫽

;

兺

i⫽3

5 a⫺

i⫺1

兺

i⫽4

9 a⫺

i⫺1

兺

j⫽1

3 a

j⫺1

兺

k⫽1

5 a

k⫺1

兺

k⫽1

兺

j⫽1

⫺

⫹

⫺

;

⫹

;

4 ⫺ 12 ⫹ 36 ⫺

;

16 ⫺ 8 ⫹ 4 ⫺

;

2 ⫹ 8 ⫹ 32 ⫹

;

2 ⫹ 6 ⫹ 18 ⫹

⫽⫺1, r ⫽⫺

;

⫽ 8, r ⫽

;

⫽ 12, r ⫽

;

⫽ 96, r ⫽

;

⫽ 2, r ⫽⫺3;

⫽ 8, r ⫽⫺2;

9–33 Section 9.3 Geometric Sequences 793

College Algebra Graphs & Models—

89. ﬁnd S

90. ﬁnd S

91. ﬁnd S

92. ﬁnd S

Determine whether the inﬁnite geometric series has a

ﬁnite sum. If so, ﬁnd the limiting value.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

105. 106.

107. 108.

兺

k⫽1

12 a

兺

j⫽1

9 a⫺

兺

i⫽1

5 a

兺

k⫽1

0.63 ⫹ 0.0063 ⫹ 0.000063 ⫹

0.3 ⫹ 0.03 ⫹ 0.003 ⫹

10 ⫺ 5 ⫹

⫺

⫹

6 ⫺ 3 ⫹

⫺

⫹

⫺49 ⫹ 1⫺72⫹ 1⫺

2⫹

6 ⫹ 3 ⫹

⫹

10 ⫹ 2 ⫹

⫹

25 ⫹ 10 ⫹ 4 ⫹

⫹

4 ⫹ 8 ⫹ 16 ⫹ 32 ⫹

3 ⫹ 6 ⫹ 12 ⫹ 24 ⫹

36 ⫹ 24 ⫹ 16 ⫹

9 ⫹ 3 ⫹ 1 ⫹

⫽ 3, a

⫽ 913;

⫽ 212, a

⫽ 8;

⫽

, a

⫽

;

⫽

, a

⫽

;

䊳

WORKING WITH FORMULAS

109. Sum of the cubes of the ﬁrst n natural numbers:

Compute using the

formula given. Then conﬁrm the result by direct

calculation.

⫹ 2

⫹ 3

⫹

⫹ 8

ⴝ

1n ⴙ 12

110. Student loan payment:

If P dollars is borrowed at an annual interest rate r

with interest compounded annually, the amount of

money to be paid back after n years is given by the

indicated formula. Find the total amount of money

that the student must repay to clear the loan, if

$8000 is borrowed at 4.5% interest and the loan is

paid back in 10 yr.

ⴝ P11 ⴙ r2

cob19545_ch09_783-796.qxd 11/9/10 9:11 PM Page 793

794 CHAPTER 9 Additional Topics in Algebra 9–34

College Algebra Graphs & Models—

䊳

APPLICATIONS

Write the nth term formula for each application, then

solve.

111. Pendulum movement: On each swing, a pendulum

travels only 80% as far as it did on the previous swing.

If the ﬁrst swing is 24 ft, how far does the pendulum

travel on the 7th swing? What total distance is

traveled before the pendulum comes to rest?

112. Tire swings: Ernesto is swinging to and fro on his

backyard tire swing. Using his legs and body, he

pumps each swing until reaching a maximum

height, then suddenly relaxes until the swing

comes to a stop. With each swing, Ernesto travels

75% as far as he did on the previous swing. If the

ﬁrst arc (or swing) is 30 ft, ﬁnd the distance

Ernesto travels on the 5th arc. What total distance

will he travel before coming to rest?

Identify the inaugural term and write the nth term

formula for each application, then solve.

113. Depreciation—automobiles: A certain new SUV

depreciates in value about 20% per year (meaning

it holds 80% of its value each year). If the SUV is

purchased for $46,000, how much is it worth 4 yr

later? How many years until its value is less than

$5000?

114. Depreciation—business equipment:A new

photocopier under heavy use will depreciate about

25% per year (meaning it holds 75% of its value

each year). If the copier is purchased for $7000,

how much is it worth 4 yr later? How many years

until its value is less than $1246?

115. Equipment aging—industrial oil pumps: Tests

have shown that the pumping power of a heavy-

duty oil pump decreases by 3% per month. If the

pump can move 160 gallons per minute (gpm) new,

how many gpm can the pump move 8 months later?

If the pumping rate falls below 118 gpm, the pump

must be replaced. How many months until this

pump is replaced?

116. Equipment aging—lumber production: At the

local mill, a certain type of saw blade can saw

approximately 2 log-feet/sec when it is new. As

time goes on, the blade becomes worn, and loses

6% of its cutting speed each week. How many log-

feet/sec can the saw blade cut after 6 weeks? If the

cutting speed falls below 1.2 log-feet/sec, the blade

must be replaced. During what week of operation

will this blade be replaced?

117. Population growth—United States: At the

beginning of the year 2000, the population of the

United States was approximately 277 million. If

the population is growing at a rate of 2.3% per

year, what was the population in 2010, 10 yr later?

118. Population growth—space colony: The

population of the Zeta Colony on Mars is 1000

people. Determine the population of the Colony

20 yr from now, if the population is growing at a

constant rate of 5% per year.

119. Creating a vacuum: To create a vacuum, a hand

pump is used to remove the air from an air-tight

cube with a volume of 462 in

. With each stroke of

the pump, two-ﬁfths of the air that remains in the

cube is removed. How much air remains inside

after the 5th stroke? How many strokes are

required to remove all but 12.9 in

of the air?

120. Atmospheric pressure: In 1654, scientist Otto Von

Guericke performed his famous demonstration of

atmospheric pressure and the strength of a vacuum

in front of Emperor Ferdinand III of Hungary.

After joining two hemispheres with mating rims,

he used a vacuum pump to remove all of the air

from the sphere formed. He then attached a team of

15 horses to each hemisphere and despite their

efforts, they could not pull the hemispheres apart.

If the sphere held a volume of 4200 in

of air and

one-tenth of the remaining air was removed with

each stroke of the pump, how much air was still in

the sphere after the 11th stroke? How many strokes

were required to remove 85% of the air?

121. Treating swimming pools: In preparation for the

summer swim season, chlorine is added to

swimming pools to control algae and bacteria.

However, careful measurements must be taken as

levels above 5 ppm (parts per million) can be

highly irritating to the eyes and throat, while levels

below 1 ppm will be ineffective (3.0 to 3.5 ppm is

ideal). In addition, the water must be treated daily

since within a 24-hr period, about 25% of the

chlorine will dissipate into the air. If the chlorine

cob19545_ch09_783-796.qxd 11/9/10 9:12 PM Page 794

level in a swimming pool is 8 ppm after its initial

treatment, how many days should the County Pool

Supervisor wait before opening it up to the public?

If left untreated, how many days until the chlorine

level drops below 1 ppm?

122. Venting landﬁll gases: The gases created from the

decomposition of waste in landﬁlls must be

carefully managed, as their release can cause

terrible odors, harm the landﬁll structure, damage

vegetation, or even cause an explosion. Suppose

the accumulated volume of gas is 50,000 ft

, and

civil engineers are able to vent 2.5% of this gas

into the atmosphere daily. What volume of gas

remains after 21 days? How many days until the

volume of gas drops below 10,000 ft

123. Population growth—bacteria: A biologist ﬁnds

that the population of a certain type of bacteria

doubles each half-hour. If an initial culture has

50 bacteria, what is the population after 5 hr? How

long will it take for the number of bacteria to reach

204,800?

9–35 Section 9.3 Geometric Sequences 795

College Algebra Graphs & Models—

124. Population growth—boom towns: Suppose the

population of a “boom town” in the old west

doubled every 2 months after gold was discovered.

If the initial population was 219, what was the

population 8 months later? How many months until

the population exceeded 28,000?

125. Elastic rebound—super balls: Megan discovers

that a rubber ball dropped from a height of 2 m

rebounds four-ﬁfths of the distance it has

previously fallen. How high does it rebound on the

7th bounce? How far does the ball travel before

coming to rest?

126. Elastic rebound—computer animation: The

screen saver on my computer is programmed to

send a colored ball vertically down the middle of

the screen so that it rebounds 95% of the distance it

last traversed. If the ball always begins at the top

and the screen is 36 cm tall, how high does the ball

bounce on its 8th rebound? How far does the ball

travel before coming to rest (and a new screen

saver starts)?

䊳

EXTENDING THE CONCEPT

127. A standard piece of typing paper is approximately

0.001 in. thick. Suppose you were able to fold this

piece of paper in half 26 times. How thick would

the result be? As tall as a hare, as tall as a hen, as

tall as a horse, as tall as a house, or over 1 mi high?

Find the actual height by computing the 27th term

of a geometric sequence. Discuss what you ﬁnd.

128. As part of a science experiment, identical rubber balls

are dropped from a given height onto these surfaces:

slate, cement, and asphalt. When dropped onto slate,

the ball rebounds 80% of the height from which it last

fell. Onto cement the ﬁgure is 75% and onto asphalt

the ﬁgure is 70%. The ball is dropped from 130 m

onto the slate, 175 m onto the cement, and 200 m

onto the asphalt. Which ball has traveled the shortest

total distance at the time of the fourth bounce? Which

ball will travel farthest before coming to rest?

129. Consider the following situation. A person is hired

at a salary of $40,000 per year, with a guaranteed

raise of $1750 per year. At the same time, inﬂation

is running about 4% per year. How many years

until this person’s salary is overtaken and eaten up

by the actual cost of living?

130. Find an alternative formula for the sum

that does not use the sigma notation.

131. Verify the following statements:

a. If a

, a

,, a

is a geometric sequence with

r and a

greater than zero, then log a

, log a

log a

, , log a

is an arithmetic sequence.

b. If a

, a

,, a

is an arithmetic sequence,

then , is a geometric

sequence.

, p ,10

, 10

⫽

兺

k⫽1

log k,

䊳

MAINTAINING YOUR SKILLS

132. (3.2) Find the zeroes of f using the quadratic

formula:

133. (R.5) Solve for x:

⫺ 3x ⫺ 10

⫺

x ⫺ 5

⫽

x ⫹ 2

1x2⫽ x

⫹ 5x ⫹ 9.

134. (4.5) Graph the rational function:

135. (5.6) Given the logistic function shown, ﬁnd p(50),

p(75), p(100), and p(150):

p1t2⫽

4200

1 ⫹ 10e

⫺0.055t

h1x2⫽

x ⫺ 1

cob19545_ch09_783-796.qxd 11/9/10 9:12 PM Page 795

College Algebra Graphs & Models—

9.4 Mathematical Induction

Since middle school (or even before) we have accepted that, “The product of two neg-

ative numbers is a positive number.” But have you ever been asked to prove it? It’s not

as easy as it seems. We may think of several patterns that yield the result, analogies that

indicate its truth, or even number line illustrations that lead us to believe the statement.

But most of us have never seen a proof (see www.mhhe.com/coburn). In this section,

we introduce one of mathematics’ most powerful tools for proving a statement, called

proof by induction.

A. Subscript Notation and Function Notation

One of the challenges in understanding a proof by induction is working with the nota-

tion. Earlier in the chapter, we introduced subscript notation as an alternative to func-

tion notation, since it is more commonly used when the functions are deﬁned by a

sequence. But regardless of the notation used, the functions can still be simpliﬁed,

evaluated, composed, and even graphed. Consider the function and the

sequence deﬁned by Both can be evaluated and graphed, with the only

difference being that f(x) is continuous with domain , while a

is discrete (made

up of distinct points) with domain .

EXAMPLE 1

䊳

Using Subscript Notation for a Composition

For and ﬁnd and

Solution

䊳

Now try Exercises 7 through 18

䊳

No matter which notation is used, every occurrence of the input variable is

replaced by the new value or expression indicated by the composition.

⫽ 3k

⫹ 6k ⫹ 2 ⫽ 3k

⫹ 6k ⫹ 2

⫽ 31k

⫹ 2k ⫹ 12⫺ 1 ⫽ 31k

⫹ 2k ⫹ 12⫺ 1

k⫹1

⫽ 31k ⫹ 12

⫺ 1 f 1k ⫹ 12⫽ 31k ⫹ 12

⫺ 1

k⫹1

.f 1k ⫹ 12a

⫽ 3n

⫺ 1,f 1x2⫽ 3x

⫺ 1

n 僆⺞

x 僆⺢

⫽ 3n

⫺ 1.

1x2⫽ 3x

⫺ 1

LEARNING OBJECTIVES

In Section 9.4 you will see

how we can:

A. Use subscript notation to

evaluate and compose

functions

B. Apply the principle of

mathematical induction

to sum formulas involving

natural numbers

C. Apply the principle of

mathematical induction

to general statements

involving natural numbers

A. You’ve just seen how

we can use subscript notation

to evaluate and compose

functions

B. Mathematical Induction Applied to Sums

Consider the sum of odd numbers . The sum of

the ﬁrst four terms is or If we now add a

(the next

term in line), would we get the same answer as if we had simply computed S

Common sense would say, “Yes!” since and

✓. In diagram form, we haveS

⫹ a

⫽ 16 ⫹ 9 ⫽ 25

⫽ 1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫽ 25

⫽ 16.1 ⫹ 3 ⫹ 5 ⫹ 7 ⫽ 16,

1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫹ 11 ⫹ 13 ⫹

add next term to

1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫹ 11 ⫹ 13 ⫹ 15 ⫹

⫽ 9

sum of 4 terms

sum of 5 terms

Our goal is to develop this same degree of clarity in the notational scheme of

things. For a given series, if we ﬁnd the kth partial sum S

(shown next) and then add

the next term would we get the same answer if we had simply computed

In other words, is true?S

⫹ a

k⫹1

⫽ S

k⫹1

796 9–36

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9–37 Section 9.4 Mathematical Induction 797

College Algebra Graphs & Models—

sum of

terms

add next term

k1



k1



k1



n–1



k1

sum of termsk  1

Now, let’s return to the sum This is an arith-

metic series with and nth term Using the sum formula

for an arithmetic sequence, an alternative formula for this sum can be established.

summation formula for an arithmetic sequence

substitute 1 for

and 2

ⴚ1 for

simplify

result

This shows that the sum of the ﬁrst n positive odd integers is given by As a

check we compute and compare to ✓.

We also note and showing

For more on this relationship, see Exercises 19 through 24. While it may seem

simplistic now, showing and (in general) is a critical

component of a proof by induction.

Unfortunately, general summation formulas for many sequences cannot be estab-

lished from known formulas. In addition, just because a formula works for the ﬁrst few

values of n, we cannot assume that it will hold true for all values of n (there are inﬁ-

nitely many). As an illustration, the formula yields a prime number

for every natural number n from 1 to 40, but fails to yield a prime for . This

helps demonstrate the need for a more conclusive proof, particularly when a relation-

ship appears to be true, and can be “veriﬁed” in a ﬁnite number of cases, but whether

it is true in all cases remains in doubt.

Proof by induction is based

on a relatively simple idea. To

help understand how it works,

consider n relay stations that are

used to transport electricity from a

generating plant to a distant city.

If we know the generating plant is

operating, and if we assume that

the kth relay station (any station

in the series) is making the trans-

fer to the station (the

next station in the series), then we’re sure the city will have electricity.

This idea can be applied mathematically as follows. Consider the statement, “The

sum of the ﬁrst n positive even integers is ” In other words,

We can certainly verify the statement for the

ﬁrst few even numbers:

The ﬁrst even number is 2 and

The sum of the ﬁrst two even numbers is and

The sum of the ﬁrst three even numbers is

and

The sum of the ﬁrst four even numbers is

and 142

 4  20p2  4  6  8  20

132

 3  12p2  4  6  12

122

 2  6p2  4  6

112

 1  2p

2  4  6  8 

 2n  n

 n.

1k  12st

n  41

 n

 n  41

 a

k1

 S

k1

 a

 S

 a

 25  11  36,S

 6

 36,

 5

 25S

 1  3  5  7  9  25

 n



n12n2



n11  2n ⴚ 12



n1a

 a

 2n  1.a

 1, d  2,

1  3  5  7 

 12n  12.

Generating plant

kth

relay

(k  1)st

relay

WORTHY OF NOTE

No matter how distant the city or

how many relay stations are

involved, if the generating plant is

working and the kth station relays

to the (k  1)st station, the city will

get its power.

cob19545_ch09_797-805.qxd 11/9/10 9:38 PM Page 797

While we could continue this process for a very long time (or even use a computer),

no ﬁnite number of checks can prove a statement is universally true. To prove the state-

ment true for all positive integers, we use a reasoning similar to that applied in the relay

stations example. If we are sure the formula works for (the generating station is op-

erating), and if the truth of implies that is true [the kth relay station is

transferring electricity to the station], then the statement is true for all n(the city

will get its electricity). The case where is called the base case of an inductive proof,

and the assumption that the formula is true for is called the induction hypothesis.

When the induction hypothesis is applied to a sum formula, we attempt to show that

Since k and are arbitrary, the statement must be true for all n.

Mathematical Induction Applied to Sums

Let S

be a sum formula involving positive integers.

If 1. S

is true, and

2. the truth of S

implies that is true,

then S

must be true for all positive integers n.

Both parts 1 and 2 must be veriﬁed for the proof to be complete. Since the process

requires the terms and we will usually compute these ﬁrst.

EXAMPLE 2

䊳

Proving a Statement Using Mathematical Induction

Use induction to prove that the sum of the ﬁrst n perfect squares is given by

Solution

䊳

Given and the needed components are

For and

1. Show S

is true for

sum formula

base case: n

 1

✓ result checks, the ﬁrst term is 1

2. Assume S

is true,

induction hypothesis:

is true

and use it to show the truth of follows. That is,

Working with the left-hand side, we have

induction hypothesis: substitute

for

common denominator



k1k  1212k  12 61k  12

1  4  9  16  25 

 k

k1k  1212k  12



k1k  1212k  12

 1k  12

1  4  9  16 

 k

 1k  12

k1

1  4  9  16 

 k

 1k  12



1k  121k  2212k  32

k1

1  4  9  16 

 k



k1k  1212k  12

 1



1122132



n1n  1212n  12

n  1.

k1



1k  121k  2212k  32



n1n  1212n  12

: S



k1k  1212k  12

k1

 1k  12

 n

: a

 k



n1n  1212n  12

 n

1  4  9  16  25 

 n



n1n  1212n  12

k1

, a

k1

k  1S

 a

k1

 S

k1

n  k

n  1

1k  12st

n  k  1n  k

n  1

798 CHAPTER 9 Additional Topics in Algebra 9–38

College Algebra Graphs & Models—

WORTHY OF NOTE

To satisfy our ﬁnite minds, it might

help to show that S

is true for the

ﬁrst few cases, prior to extending

the ideas to the inﬁnite case.

⎫

⎪

⎬

⎪

⎭

⎫

⎪

⎬

⎪

⎭

⎫

⎬

⎭

cob19545_ch09_797-805.qxd 12/17/10 12:54 AM Page 798

9–39 Section 9.4 Mathematical Induction 799

College Algebra Graphs & Models—

factor out

 1

multiply and combine terms

factor the trinomial, result is

Since the truth of follows from S

, the formula is true for all n.

Now try Exercises 27 through 38

䊳

C. The General Principle of Mathematical Induction

Proof by induction can be used to verify many other kinds of relationships involving a

natural number n. In this regard, the basic principles remain the same but are stated

more broadly. Rather than using S

to represent a sum, we will use P

to represent any

proposed statement or relationship we might wish to verify. This broadens the scope

of the proof and makes it more widely applicable, while maintaining its connection to

the sum formulas veriﬁed earlier.

The General Principle of Mathematical Induction

Let P

be a statement involving natural numbers.

If 1. P

is true, and

2. the truth of P

implies that is also true

then P

must be true for all natural numbers n.

k1



1k  121k  2212k  32



1k  1232k

 7k  64



1k  123k12k  12 61k  124

B. You’ve just seen how

we can apply the principle of

mathematical induction to

sum formulas involving

natural numbers

EXAMPLE 3

䊳

Proving a Statement Using the General Principle of Mathematical Induction

Use the general principle of mathematical induction to show the statement P

true for all natural numbers n.

Solution

䊳

The statement P

is deﬁned as This means that P

is represented by

and by

1. Show P

is true for

given statement

base case:

✓ true

Although not a part of the formal proof, a table of values can help to illustrate

the relationship we’re trying to establish. It appears that the statement is true.

2  2

n  1 2

 1  1P

 n  1P

n  1:

k1

 k  2.P

k1

 k  1

 n  1.

: 2

 n  1

n 123 4 5

2 4 8 16 32

n  1234 5 6

2. Assume that P

is true.

induction hypothesis

and use it to show the truth of . That is,

Begin by working with the left-hand side of the inequality, .2

k1

 k  2

k1

 1k  12 1

k1

 k  1

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