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
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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
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, 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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CHAPTER 9 Additional Topics in Algebra
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?
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 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
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 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?
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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.
MAINTAINING YOUR SKILLS
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
add next term akϩ1
>
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