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E. Applications Involving Geometric Sequences and Series

# E. Applications Involving Geometric Sequences and Series

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CHAPTER 9 Additional Topics in Algebra

Solution

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

geometric sequence with a1 ϭ 2 and r ϭ 0.9.

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

so on. For the 8th term we have:

an ϭ a1rnϪ1

a8 ϭ 210.92 8Ϫ1

Ϸ 0.957

n th term formula

substitute 8 for n, 2 for a1, and 0.9 for r

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 S8.

Sn ϭ

S8 ϭ

a1 11 Ϫ rn 2

1Ϫr

211 Ϫ 0.98 2

1 Ϫ 0.9

Ϸ 11.4

Figure 9.48

Figure 9.49

n th partial sum formula

substitute 2 for a1, 0.9 for r,

and 8 for n

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 0.5 ϭ 210.92 nϪ1. This yields

0.25 ϭ 10.92 nϪ1

ln 0.25 ϭ 1n Ϫ 12ln 0.9

ln 0.25

ϩ1ϭn

ln 0.9

14.16 Ϸ n

Figure 9.50

divide by 2

take the natural log,

apply power property

solve for n

(exact form)

solve for n

(approximate form)

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

Figure 9.51

to rest, we consider the related infinite

geometric series, with a1 ϭ 2 and r ϭ 0.9.

a1

infinite sum formula

Sq ϭ

1Ϫr

2

substitute 2 for a1 and 0.9 for r

Sq ϭ

1 Ϫ 0.9

result

ϭ 20

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 S15 and S150 are shown in Figure 9.51.

Now try Exercises 111 and 112

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

application will use the preliminary or inaugural term a0, 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.

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Section 9.3 Geometric Sequences

EXAMPLE 10

791

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

The efficiency of a new drill bit (prior to use) is given as a0 ϭ 6 cm/sec. Since

the bit loses 4% ϭ 0.04 of its efficiency per day, it maintains 96% ϭ 0.96 of its

efficiency, showing that after 1 day of use a1 ϭ 0.96162 ϭ 5.76. This means the nth

term formula will be an ϭ 5.7610.962 nϪ1.

a. At the end of day 5 we have

a5 ϭ 5.7610.962 5Ϫ1

ϭ 5.7610.962 4

Ϸ 4.9

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 an with 3.6 and solve for n.

E. You’ve just seen how

we can solve application

problems involving geometric

sequences and series

3.6 ϭ 5.7610.962 nϪ1

0.625 ϭ 0.96nϪ1

ln 0.625 ϭ ln 0.96nϪ1

ln 0.625 ϭ 1n Ϫ 12 ln 0.96

ln 0.625

ϭnϪ1

ln 0.96

ln 0.625

ϩ1ϭn

ln 0.96

12.5 Ϸ n

substitute 3.6 for an

divide

take the natural log of both sides

power property

divide

solve for n (exact form)

solution (approximate form)

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

Now try Exercises 113 through 126

9.3 EXERCISES

CONCEPTS AND VOCABULARY

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

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

an ϭ

, for any n Ն 1.

4. For the general sequence a1, a2, a3, p , ak, p , the

fifth partial sum is given by S5 ϭ

.

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?

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CHAPTER 9 Additional Topics in Algebra

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, p

14. 12, 0.12, 0.0012, 0.000012, p

15. Ϫ1, 3, Ϫ12, 60, Ϫ360, p

16. Ϫ 23, 2, Ϫ8, 40, Ϫ240, p

47. a1 ϭ 9, an ϭ 729, r ϭ 3

48. a1 ϭ 1, an ϭ Ϫ128, r ϭ Ϫ2

1

, r ϭ 12

49. a1 ϭ 16, an ϭ 64

1

, r ϭ 12

50. a1 ϭ 4, an ϭ 512

p

51. a1 ϭ Ϫ1, an ϭ Ϫ1296, r ϭ 16

18. Ϫ36, 24, Ϫ16, 32

3,p

52. a1 ϭ 2, an ϭ 1458, r ϭ Ϫ 13

1

19. 12, 14, 18, 16

,p

53. 2, Ϫ6, 18, Ϫ54, p , Ϫ4374

8 16

20. 23, 49, 27

, 81, p

54. 3, Ϫ6, 12, Ϫ24, p , Ϫ6144

12 48 192

21. 3, , 2 , 3 , p

x x x

55. 64, 3212, 32, 1612, p , 1

56. 243, 81 13, 81, 27 13, p , 1

10 20 40

22. 5, , 2 , 3 , p

a a a

57. 38, Ϫ34, 32, Ϫ3, p , 96

5 5

, 9, Ϫ53, Ϫ5, p , Ϫ135

58. Ϫ27

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

24. Ϫ120, Ϫ60, Ϫ20, Ϫ5, Ϫ1, p

Write the first four terms of the sequence, given a1 and r.

25. a1 ϭ 5, r ϭ 2

27. a1 ϭ Ϫ6, r ϭ

26. a1 ϭ 2, r ϭ Ϫ4

Ϫ12

28. a1 ϭ 23, r ϭ 15

29. a1 ϭ 4, r ϭ 13

30. a1 ϭ 15, r ϭ 15

31. a1 ϭ 0.1, r ϭ 0.1

32. a1 ϭ 0.024, r ϭ 0.01

Write the expression for the nth term, then find the

indicated term for each sequence.

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, p

60. 2, 2 25, 10, 10 25, 50, 50 15, p

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

33. a1 ϭ Ϫ24, r ϭ 12; find a7

36. a1 ϭ

1, 12, 2, p

Find the number of terms in each sequence.

13. 3, 0.3, 0.03, 0.003, p

3

20 ,

42. 625, 125, 25, 5, 1, p

46. 0.5, Ϫ0.35, 0.245, Ϫ0.1715, p

12. Ϫ13, Ϫ9, Ϫ5, Ϫ1, 3, p

35. a1 ϭ

12

2 ,

40. Ϫ78, 74, Ϫ72, 7, Ϫ14, p

45. 0.2, 0.08, 0.032, 0.0128, p

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

1

Ϫ20

,

Ϫ19, 13, Ϫ1, 3, p

44. 3613, 36, 1213, 12, 4 13, p

10. 128, Ϫ32, 8, Ϫ2, p

34. a1 ϭ 48, r ϭ

1

27 ,

43. 12,

9. 3, Ϫ6, 12, Ϫ24, 48, p

17. 25, 10, 4,

39.

41. 729, 243, 81, 27, 9, p

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

8

5,

Identify a1 and r, then write the expression for the nth

term an ‫ ؍‬a1rn؊1 and use it to find a6, a10, and a12.

Ϫ13;

find a6

r ϭ Ϫ5; find a4

r ϭ 4; find a5

62.

1 1 1 2 5

, , , , , 1, p

6 3 2 3 6

Find the common ratio r and the value of a1 using the

information given (assume r 7 0).

37. a1 ϭ 2, r ϭ 12; find a7

63. a3 ϭ 324, a7 ϭ 64

64. a5 ϭ 6, a9 ϭ 486

38. a1 ϭ 13, r ϭ 13; find a8

65. a4 ϭ

66. a2 ϭ 16

81 , a5 ϭ

4

9,

a8 ϭ

9

4

67. a4 ϭ 32

3 , a8 ϭ 54

2

3

68. a3 ϭ 16

25 , a7 ϭ 25

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Section 9.3 Geometric Sequences

Find the partial sum indicated.

9

89. a3 ϭ 49, a7 ϭ 64

; find S6

69. a1 ϭ 8, r ϭ Ϫ2; find S12

2

90. a2 ϭ 16

81 , a5 ϭ 3 ; find S8

70. a1 ϭ 2, r ϭ Ϫ3; find S8

91. a3 ϭ 2 12, a6 ϭ 8; find S7

71. a1 ϭ 96, r ϭ 13; find S5

92. a2 ϭ 3, a5 ϭ 9 13; find S7

72. a1 ϭ 12, r ϭ

73. a1 ϭ 8, r ϭ

1

2;

3

2;

find S8

Determine whether the infinite geometric series has a

finite sum. If so, find the limiting value.

find S7

74. a1 ϭ Ϫ1, r ϭ Ϫ32; find S10

75. 2 ϩ 6 ϩ 18 ϩ p ; find S6

93. 9 ϩ 3 ϩ 1 ϩ p

76. 2 ϩ 8 ϩ 32 ϩ p ; find S7

77. 16 Ϫ 8 ϩ 4 Ϫ p ; find S8

95. 3 ϩ 6 ϩ 12 ϩ 24 ϩ p

96. 4 ϩ 8 ϩ 16 ϩ 32 ϩ p

78. 4 Ϫ 12 ϩ 36 Ϫ p ; find S8

1

79. 43 ϩ 29 ϩ 27

ϩ p ; find S9

97. 25 ϩ 10 ϩ 4 ϩ 85 ϩ p

2

98. 10 ϩ 2 ϩ 25 ϩ 25

ϩp

80.

1

18

94. 36 ϩ 24 ϩ 16 ϩ p

Ϫ 16 ϩ 12 Ϫ p ; find S7

99. 6 ϩ 3 ϩ 32 ϩ 34 ϩ p

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.

5

81.

͚4

10

j

82.

͚2

k

jϭ1

kϭ1

8

7

2 kϪ1

83.

5a b

3

kϭ1

1 jϪ1

84.

3a b

5

jϭ1

͚

͚

10

1 iϪ1

85.

9 aϪ b

2

iϭ4

1 iϪ1

86.

5 aϪ b

4

iϭ3

100. Ϫ49 ϩ 1Ϫ72 ϩ 1Ϫ17 2 ϩ p

101. 6 Ϫ 3 ϩ 32 Ϫ 34 ϩ p

102. 10 Ϫ 5 ϩ 52 Ϫ 54 ϩ p

103. 0.3 ϩ 0.03 ϩ 0.003 ϩ p

104. 0.63 ϩ 0.0063 ϩ 0.000063 ϩ p

105.

3 2 k

a b

kϭ1 4 3

͚

106.

1 i

5a b

2

iϭ1

107.

5 j

9 aϪ b

4

jϭ1

108.

4 k

12 a b

3

kϭ1

q

8

͚

793

͚

q

q

͚

͚

q

͚

Find the indicated partial sum using the information

given. Write all results in simplest form.

87. a2 ϭ Ϫ5, a5 ϭ

1

25 ;

find S5

88. a3 ϭ 1, a6 ϭ Ϫ27; find S6

WORKING WITH FORMULAS

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

n2 1n ؉ 12 2

Sn ‫؍‬

4

3

Compute 1 ϩ 23 ϩ 33 ϩ p ϩ 83 using the

formula given. Then confirm the result by direct

calculation.

110. Student loan payment: An ‫ ؍‬P11 ؉ r2 n

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.

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9–34

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?

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 heavyduty 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 logfeet/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

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 in3. 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 in3 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 in3 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

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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 ft3, 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 ft3?

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?

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)?

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?

795

Section 9.3 Geometric Sequences

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

n

Sn ϭ

͚ log k, that does not use the sigma notation.

kϭ1

131. Verify the following statements:

a. If a1, a2, a3, p , an is a geometric sequence with

r and a1 greater than zero, then log a1, log a2,

log a3, p , log an is an arithmetic sequence.

b. If a1, a2, a3, p , an is an arithmetic sequence,

then 10a1, 10a2, p , 10an, is a geometric

sequence.

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

formula: f 1x2 ϭ x2 ϩ 5x ϩ 9.

133. (R.5) Solve for x:

4

3

1

Ϫ

ϭ

x

Ϫ

5

x

ϩ

2

x Ϫ 3x Ϫ 10

2

134. (4.5) Graph the rational function: h1x2 ϭ

x2

xϪ1

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

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

4200

p1t2 ϭ

1 ϩ 10eϪ0.055t

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9.4

Mathematical Induction

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

EXAMPLE 1

Solution

A. You’ve just seen how

we can use subscript notation

to evaluate and compose

functions

Since middle school (or even before) we have accepted that, “The product of two negative 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 notation. Earlier in the chapter, we introduced subscript notation as an alternative to function 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 f 1x2 ϭ 3x2 Ϫ 1 and the

sequence defined by an ϭ 3n2 Ϫ 1. Both can be evaluated and graphed, with the only

difference being that f(x) is continuous with domain x ʦ ‫ޒ‬, while an is discrete (made

up of distinct points) with domain n ʦ ‫ގ‬.

Using Subscript Notation for a Composition

For f 1x2 ϭ 3x2 Ϫ 1 and an ϭ 3n2 Ϫ 1, find f 1k ϩ 12 and akϩ1.

f 1k ϩ 12 ϭ 31k ϩ 12 2 Ϫ 1

ϭ 31k2 ϩ 2k ϩ 12 Ϫ 1

ϭ 3k2 ϩ 6k ϩ 2

akϩ1 ϭ 31k ϩ 12 2 Ϫ 1

ϭ 31k2 ϩ 2k ϩ 12 Ϫ 1

ϭ 3k2 ϩ 6k ϩ 2

Now try Exercises 7 through 18

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

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

B. Mathematical Induction Applied to Sums

Consider the sum of odd numbers 1 ϩ 3 ϩ 5 ϩ 7 ϩ 9 ϩ 11 ϩ 13 ϩ p . The sum of

the first four terms is 1 ϩ 3 ϩ 5 ϩ 7 ϭ 16, or S4 ϭ 16. If we now add a5 (the next

term in line), would we get the same answer as if we had simply computed S5?

Common sense would say, “Yes!” since S5 ϭ 1 ϩ 3 ϩ 5 ϩ 7 ϩ 9 ϭ 25 and

S4 ϩ a5 ϭ 16 ϩ 9 ϭ 25✓. In diagram form, we have

add next term a5 ϭ 9 to S4

>

1 ϩ 3 ϩ 5 ϩ 7 ϩ 9 ϩ 11 ϩ 13 ϩ 15 ϩ p

c

c

S4

S5

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 Sk (shown next) and then add

the next term akϩ1, would we get the same answer if we had simply computed Skϩ1?

In other words, is Sk ϩ akϩ1 ϭ Skϩ1 true?

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Section 9.4 Mathematical Induction

797

>

a1 ϩ a2 ϩ a3 ϩ p ϩ akϪ1 ϩ ak ϩ akϩ1 ϩ p ϩ an–1 ϩ an

c

c

sum of k terms

Sk

sum of k ϩ 1 terms

Skϩ1

Now, let’s return to the sum 1 ϩ 3 ϩ 5 ϩ 7 ϩ p ϩ 12n Ϫ 12. This is an arithmetic series with a1 ϭ 1, d ϭ 2, and nth term an ϭ 2n Ϫ 1. Using the sum formula

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

Sn ϭ

ϭ

ϭ

n1a1 ϩ an 2

2

n11 ϩ 2n ؊ 12

2

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 ϩ 1)st station, the city will

get its power.

substitute 1 for a1 and 2n؊1 for an

n12n2

ϭ n2

WORTHY OF NOTE

summation formula for an arithmetic sequence

2

simplify

result

This shows that the sum of the first n positive odd integers is given by Sn ϭ n2. As a

check we compute S5 ϭ 1 ϩ 3 ϩ 5 ϩ 7 ϩ 9 ϭ 25 and compare to S5 ϭ 52 ϭ 25✓.

We also note S6 ϭ 62 ϭ 36, and S5 ϩ a6 ϭ 25 ϩ 11 ϭ 36, showing S6 ϭ S5 ϩ a6.

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

simplistic now, showing S5 ϩ a6 ϭ S6 and Sk ϩ akϩ1 ϭ Skϩ1 (in general) is a critical

component of a proof by induction.

Unfortunately, general summation formulas for many sequences cannot be established 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 infinitely many). As an illustration, the formula an ϭ n2 Ϫ n ϩ 41 yields a prime number

for every natural number n from 1 to 40, but fails to yield a prime for n ϭ 41. This

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

(k ϩ 1)st

kth

Generating plant

in the series) is making the transrelay

relay

fer to the 1k ϩ 12st 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 n2 ϩ n.” In other words,

2 ϩ 4 ϩ 6 ϩ 8 ϩ p ϩ 2n ϭ n2 ϩ n. We can certainly verify the statement for the

first few even numbers:

112 2 ϩ 1 ϭ 2

The first even number is 2 and p

122 2 ϩ 2 ϭ 6

The sum of the first two even numbers is 2 ϩ 4 ϭ 6 and p

The sum of the first three even numbers is

132 2 ϩ 3 ϭ 12

2 ϩ 4 ϩ 6 ϭ 12 and p

The sum of the first four even numbers is

2 ϩ 4 ϩ 6 ϩ 8 ϭ 20 and p

142 2 ϩ 4 ϭ 20

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E. Applications Involving Geometric Sequences and Series

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