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Appendix D
D
■
Mathematical Induction
D1
Mathematical Induction
■ Use mathematical induction to prove a formula.
■ Find a sum of powers of integers.
■ Find a formula for a finite sum.
■ Use finite differences to find a linear or quadratic model.
Introduction
In this section, you will study a form of mathematical proof called mathematical
induction. It is important that you clearly see the logical need for it, so take a look at
the problem below.
⫽1
⫽1
⫽1
⫽1
S5 ⫽ 1
S1
S2
S3
S4
⫽ 12
⫹ 3 ⫽ 22
⫹ 3 ⫹ 5 ⫽ 32
⫹ 3 ⫹ 5 ⫹ 7 ⫽ 42
⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫽ 52
Judging from the pattern formed by these first five sums, it appears that the sum of the
first n odd integers is
Sn ⫽ 1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ 9 ⫹ . . . ⫹ 共2n ⫺ 1兲 ⫽ n2.
Although this particular formula is valid, it is important for you to see that recognizing
a pattern and then simply jumping to the conclusion that the pattern must be true for all
values of n is not a logically valid method of proof. There are many examples in which
a pattern appears to be developing for small values of n and then at some point the
pattern fails. One of the most famous cases of this was the conjecture by the French
mathematician Pierre de Fermat (1601–1665), who speculated that all numbers of the
form
Fn ⫽ 22 ⫹ 1,
n
n ⫽ 0, 1, 2, . . .
are prime. For n ⫽ 0, 1, 2, 3, and 4, the conjecture is true.
F0 ⫽ 3, F1 ⫽ 5, F2 ⫽ 17, F3 ⫽ 257, F4 ⫽ 65,537
The size of the next Fermat number 共F5 ⫽ 4,294,967,297兲 is so great that it was
difficult for Fermat to determine whether it was prime or not. However, another
well-known mathematician, Leonhard Euler (1707–1783), later found the factorization
F5 ⫽ 4,294,967,297
⫽ 641共6,700,417兲
which proved that F5 is not prime and therefore Fermat’s conjecture was false.
Just because a rule, pattern, or formula seems to work for several values of n,
you cannot simply decide that it is valid for all values of n without going through a
legitimate proof. Mathematical induction is one method of proof.
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Mathematical Induction
The Principle of Mathematical Induction
Let Pn be a statement involving the positive integer n. If
1. P1 is true, and
2. the truth of Pk implies the truth of Pk⫹1 for every positive integer k,
then Pn must be true for all positive integers n.
It is important to recognize that both parts of the Principle of Mathematical
Induction are necessary. To apply the Principle of Mathematical Induction, you need to
be able to determine the statement Pk⫹1 for a given statement Pk.
Example 1
Using Pk to Find Pkⴙ1
Find Pk⫹1 for each statement.
a. Pk: Sk ⫽
k 2共k ⫹ 1兲2
4
b. Pk: Sk ⫽ 1 ⫹ 5 ⫹ 9 ⫹ . . . ⫹ 关4共k ⫺ 1兲 ⫺ 3兴 ⫹ 共4k ⫺ 3兲
c. Pk: 3k ≥ 2k ⫹ 1
SOLUTION
共k ⫹ 1兲2共k ⫹ 1 ⫹ 1兲2
Replace k by k ⫹ 1.
4
共k ⫹ 1兲2共k ⫹ 2兲2
Simplify.
⫽
4
b. Pk⫹1: Sk⫹1 ⫽ 1 ⫹ 5 ⫹ 9 ⫹ . . . ⫹ 再4关共k ⫹ 1) ⫺ 1] ⫺ 3冎 ⫹ 关4共k ⫹ 1兲 ⫺ 3兴
⫽ 1 ⫹ 5 ⫹ 9 ⫹ . . . ⫹ 共4k ⫺ 3兲 ⫹ 共4k ⫹ 1兲
a. Pk⫹1: Sk⫹1 ⫽
c. Pk⫹1: 3k⫹1 ≥ 2共k ⫹ 1兲 ⫹ 1
3k⫹1 ≥ 2k ⫹ 3
Checkpoint 1
Find Pk⫹1 for Pk: Sk ⫽ k共2k ⫹ 1兲.
FIGURE D.26
■
A well-known illustration used to explain why the Principle of Mathematical
Induction works is the unending line of dominoes shown in Figure D.26. If the line
actually contains infinitely many dominoes, then it is clear that you could not knock the
entire line down by knocking down only one domino at a time. However, suppose it
were true that each domino would knock down the next one as it fell. Then you could
knock them all down simply by pushing the first one and starting a chain reaction.
Mathematical induction works in the same way. If the truth of Pk implies the truth of
Pk⫹1 and if P1 is true, then the chain reaction proceeds as follows: P1 implies P2, P2
implies P3, P3 implies P4, and so on.
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Mathematical Induction
D3
Using Mathematical Induction
Example 2
Using Mathematical Induction
Use mathematical induction to prove the formula
Sn ⫽ 1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ . . . ⫹ 共2n ⫺ 1兲 ⫽ n2
for all integers n ≥ 1.
Mathematical induction consists of two distinct parts. First, you must
show that the formula is true when n ⫽ 1.
SOLUTION
1. When n ⫽ 1, the formula is valid, because
S1 ⫽ 1 ⫽ 12.
The second part of mathematical induction has two steps. The first step is to assume that
the formula is valid for some integer k. The second step is to use this assumption to
prove that the formula is valid for the next integer, k ⫹ 1.
2. Assuming that the formula
Sk ⫽ 1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ . . . ⫹ 共2k ⫺ 1兲 ⫽ k2
is true, you must show that the formula Sk⫹1 ⫽ 共k ⫹ 1兲2 is true.
STUDY TIP
When using mathematical
induction to prove a
summation formula (such as
the one in Example 2), it is
helpful to think of Sk⫹1 as
Sk⫹1 ⫽ Sk ⫹ ak⫹1, where
ak⫹1 is the 共k ⫹ 1兲th term
of the original sum.
Sk⫹1 ⫽ 1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ . . . ⫹ 共2k ⫺ 1兲 ⫹ 关2共k ⫹ 1兲 ⫺ 1兴
⫽ 关1 ⫹ 3 ⫹ 5 ⫹ 7 ⫹ . . . ⫹ 共2k ⫺ 1兲兴 ⫹ 共2k ⫹ 2 ⫺ 1兲
⫽ Sk ⫹ 共2k ⫹ 1兲
Group terms to form Sk.
⫽ k2 ⫹ 2k ⫹ 1
Replace Sk by k2.
2
⫽ 共k ⫹ 1兲
Factor.
Combining the results of parts (1) and (2), you can conclude by mathematical induction
that the formula is valid for all integers n ≥ 1.
Checkpoint 2
Use mathematical induction to prove the formula
n共n ⫹ 7兲
Sn ⫽ 4 ⫹ 5 ⫹ 6 ⫹ 7 ⫹ . . . ⫹ 共n ⫹ 3兲 ⫽
2
for all integers n ≥ 1.
■
It occasionally happens that a statement involving natural numbers is not true for
the first k ⫺ 1 positive integers but is true for all values of n ≥ k. In these instances,
you use a slight variation of the Principle of Mathematical Induction in which you
verify Pk rather than P1. This variation is called the Extended Principle of
Mathematical Induction. To see the validity of this variation, note from Figure D.26
that all but the first k ⫺ 1 dominoes can be knocked down by knocking over the kth
domino. This suggests that you can prove a statement Pn to be true for n ≥ k by
showing that Pk is true and that Pk implies Pk⫹1. In Exercises 35–38 of this section, you
are asked to apply this extension of mathematical induction.
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Mathematical Induction
Example 3
Using Mathematical Induction
Use mathematical induction to prove the formula
Sn ⫽ 12 ⫹ 22 ⫹ 32 ⫹ 42 ⫹ . . . ⫹ n 2
n共n ⫹ 1兲共2n ⫹ 1兲
⫽
6
for all integers n ≥ 1.
SOLUTION
1. When n ⫽ 1, the formula is valid, because
S1 ⫽ 12 ⫽
1共1 ⫹ 1兲共2
6
⭈ 1 ⫹ 1兲 ⫽ 1共2兲共3兲.
6
2. Assuming that
k共k ⫹ 1兲共2k ⫹ 1兲
Sk ⫽ 12 ⫹ 22 ⫹ 32 ⫹ 42 ⫹ . . . ⫹ k2 ⫽
6
you must show that
Sk⫹1 ⫽
共k ⫹ 1兲共k ⫹ 1 ⫹ 1兲关2共k ⫹ 1兲 ⫹ 1兴 (k ⫹ 1兲共k ⫹ 2兲共2k ⫹ 3兲
⫽
6
6
To do this, write the following.
Sk⫹1 ⫽ Sk ⫹ ak⫹1
⫽ 共12 ⫹ 22 ⫹ 32 ⫹ 42 ⫹ . . . ⫹ k 2兲 ⫹ 共k ⫹ 1兲2
k共k ⫹ 1兲共2k ⫹ 1兲
⫽
⫹ 共k ⫹ 1兲2
By assumption
6
k共k ⫹ 1兲共2k ⫹ 1兲 ⫹ 6共k ⫹ 1兲2
⫽
Add expressions.
6
共k ⫹ 1兲关k共2k ⫹ 1兲 ⫹ 6共k ⫹ 1兲兴
⫽
Factor out 共k ⫹ 1兲.
6
共k ⫹ 1兲共2k2 ⫹ 7k ⫹ 6兲
⫽
Simplify.
6
共k ⫹ 1兲共k ⫹ 2兲共2k ⫹ 3兲
⫽
Completely factor.
6
Combining the results of parts (1) and (2), you can conclude by mathematical induction
that the formula is valid for all integers n ≥ 1.
Checkpoint 3
Use mathematical induction to prove the formula
3n共n ⫹ 1兲
Sn ⫽ 3 ⫹ 6 ⫹ 9 ⫹ 12 ⫹ . . . ⫹ 3n ⫽
2
for all integers n ⫽ 1.
■
When proving a formula by mathematical induction, the only statement that you
need to verify is P1. As a check, however, it is a good idea to try verifying some of the
other statements. For instance, in Example 3, try verifying S2 and S3.
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Mathematical Induction
D5
Sums of Powers of Integers
The formula in Example 3 is one of a collection of useful summation formulas. This
and other formulas dealing with the sums of various powers of the first n positive
integers are shown.
Sums of Powers of Integers
n共n ⫹ 1兲
1. 1 ⫹ 2 ⫹ 3 ⫹ 4 ⫹ . . . ⫹ n ⫽
2
n共n ⫹ 1兲共2n ⫹ 1兲
2. 12 ⫹ 22 ⫹ 32 ⫹ 42 ⫹ . . . ⫹ n 2 ⫽
6
n 2共n ⫹ 1兲2
3. 13 ⫹ 23 ⫹ 33 ⫹ 43 ⫹ . . . ⫹ n 3 ⫽
4
n共n ⫹ 1兲共2n ⫹ 1兲共3n 2 ⫹ 3n ⫺ 1兲
4. 14 ⫹ 24 ⫹ 34 ⫹ 44 ⫹ . . . ⫹ n 4 ⫽
30
n 2共n ⫹ 1兲2共2n 2 ⫹ 2n ⫺ 1兲
5. 15 ⫹ 25 ⫹ 35 ⫹ 45 ⫹ . . . ⫹ n 5 ⫽
12
Each of these formulas for sums can be proven by mathematical induction.
Example 4
7
Find
兺n
Finding a Sum of Powers of Integers
⫽ 13 ⫹ 23 ⫹ 33 ⫹ 43 ⫹ 53 ⫹ 63 ⫹ 73.
3
n⫽1
Using the formula for the sum of the cubes of the first n positive integers,
you obtain the following.
SOLUTION
7
兺n
3
⫽ 13 ⫹ 23 ⫹ 33 ⫹ 43 ⫹ 53 ⫹ 63 ⫹ 73
n⫽1
72共7 ⫹ 1兲2
4
49共64兲
⫽
4
⫽ 784
⫽
Check this sum by adding the numbers, as shown below.
1 ⫹ 8 ⫹ 27 ⫹ 64 ⫹ 125 ⫹ 216 ⫹ 343 ⫽ 784
Checkpoint 4
10
Find
兺n
2
n⫽1
⫽ 12 ⫹ 22 ⫹ 32 ⫹ 42 ⫹ 52 ⫹ 62 ⫹ 72 ⫹ 82 ⫹ 92 ⫹ 102.
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Mathematical Induction
Example 5
Proving an Inequality by Mathematical Induction
Prove that n < 2n for all positive integers n.
SOLUTION
1. For n ⫽ 1 and n ⫽ 2, the formula is true, because
1 < 21 and 2 < 22.
2. Assuming that
k < 2k
you need to show that k ⫹ 1 < 2k⫹1. For n ⫽ k, you have
2k⫹1 ⫽ 2共2k兲 > 2共k兲 ⫽ 2k.
By assumption
Because 2k ⫽ k ⫹ k > k ⫹ 1 for all k > 1, it follows that
2k⫹1 > 2k > k ⫹ 1
or
k ⫹ 1 < 2k⫹1.
Combining the results of parts (1) and (2), you can conclude by mathematical induction
that n < 2n for all positive integers n.
Checkpoint 5
Prove that n! > 2n for all integers n > 3.
■
Pattern Recognition
Although choosing a formula on the basis of a few observations does not guarantee the
validity of the formula, pattern recognition is important. Once you have a pattern that
you think works, you can try using mathematical induction to prove your formula.
Finding a Formula for the nth Term of a Sequence
To find a formula for the nth term of a sequence, consider these guidelines.
1. Calculate the first several terms of the sequence. It is often a good idea to write
the terms in both simplified and factored forms.
2. Try to find a recognizable pattern for the terms and write a formula for the nth
term of the sequence. This is your hypothesis or conjecture. You might try
computing one or two more terms in the sequence to test your hypothesis.
3. Use mathematical induction to prove your hypothesis.
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Example 6
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Mathematical Induction
D7
Finding a Formula for a Finite Sum
Find a formula for the finite sum.
1
1
⭈2
1
⫹
2
⭈3
⫹
1
3
⭈4
⫹
1
4
⭈5
⫹. . .⫹
1
n共n ⫹ 1兲
Begin by writing out the first few sums.
SOLUTION
1
1
1
⫽
1⭈2 2 1⫹1
1
1
4 2
2
S2 ⫽
⫹
⫽ ⫽ ⫽
1⭈2 2⭈3 6 3 2⫹1
1
1
1
9
3
3
S3 ⫽
⫹
⫹
⫽
⫽ ⫽
1 ⭈ 2 2 ⭈ 3 3 ⭈ 4 12 4 3 ⫹ 1
1
1
1
1
48 4
4
S4 ⫽
⫹
⫹
⫹
⫽
⫽ ⫽
1 ⭈ 2 2 ⭈ 3 3 ⭈ 4 4 ⭈ 5 60 5 4 ⫹ 1
S1 ⫽
⫽
From this sequence, it appears that the formula for the kth sum is
Sk ⫽
⫽
1
1
⭈2
⫹
1
2
⭈3
⫹
1
3
⭈4
⫹
1
4
⭈5
⫹. . .⫹
1
k共k ⫹ 1兲
k
.
k⫹1
To prove the validity of this hypothesis, use mathematical induction, as shown below.
Note that you have already verified the formula for n ⫽ 1, so you can begin by assuming
that the formula is valid for n ⫽ k and trying to show that it is valid for n ⫽ k ⫹ 1.
Sk⫹1 ⫽
冤 1 1⭈ 2 ⫹ 2 1⭈ 3 ⫹ 3 1⭈ 4 ⫹ 4 1⭈ 5 ⫹ . . . ⫹ k共k 1⫹ 1兲冥 ⫹ 共k ⫹ 1兲共1 k ⫹ 2兲
k
1
⫹
k ⫹ 1 共k ⫹ 1兲共k ⫹ 2兲
k共k ⫹ 2兲 ⫹ 1
⫽
共k ⫹ 1兲共k ⫹ 2兲
k 2 ⫹ 2k ⫹ 1
⫽
共k ⫹ 1兲共k ⫹ 2兲
共k ⫹ 1兲2
⫽
共k ⫹ 1兲共k ⫹ 2兲
k⫹1
⫽
k⫹2
So, the hypothesis is valid.
⫽
By assumption
Add fractions.
Distributive Property
Factor numerator.
Simplify.
Checkpoint 6
Find a formula for the finite sum.
3 ⫹ 5 ⫹ 7 ⫹ 9 ⫹ . . . ⫹ 共2n ⫹ 1兲
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Mathematical Induction
Finite Differences
The first differences of a sequence are found by subtracting consecutive terms. The
second differences are found by subtracting consecutive first differences. The first and
second differences of the sequence 3, 5, 8, 12, 17, 23, . . . are as shown.
n:
1
2
3
4
5
6
an:
3
5
8
12
17
23
First differences:
2
Second differences:
3
1
4
1
5
1
6
1
For this sequence, the second differences are all the same nonzero number. When this
happens, the sequence has a perfect quadratic model. When the first differences are all
the same, the sequence has a linear model. That is, it is arithmetic.
Example 7
Finding an Appropriate Model
Decide whether the sequence 6, 12, 26, 48, 78, 116, . . . can be represented
perfectly by a linear or a quadratic model. If so, find the model.
SOLUTION
Find the first and second differences of the sequence.
n:
1
2
3
4
5
6
an:
6
12
26
48
78
116
First differences:
6
Second differences:
14
8
22
8
30
8
38
8
Because the second differences are all the same, the sequence can be represented
perfectly by a quadratic model of the form
an ⫽ an2 ⫹ bn ⫹ c.
By substituting 1, 2, and 3 for n, you can obtain a system of three linear equations in
three variables.
a1 ⫽ a共1兲2 ⫹ b共1兲 ⫹ c ⫽ 6
a2 ⫽ a共2兲2 ⫹ b共2兲 ⫹ c ⫽ 12
a3 ⫽ a共3兲2 ⫹ b共3兲 ⫹ c ⫽ 26
Substitute 1 for n.
Substitute 2 for n.
Substitute 3 for n.
You now have a system of three equations in a, b, and c.
冦
a⫹ b⫹c⫽ 6
4a ⫹ 2b ⫹ c ⫽ 12
9a ⫹ 3b ⫹ c ⫽ 26
Equation 1
Equation 2
Equation 3
Using the techniques discussed in Chapter 5, you can find that the solution of this
system is
a ⫽ 4, b ⫽ ⫺6, and c ⫽ 8.
So, the quadratic model is
an ⫽ 4n2 ⫺ 6n ⫹ 8.
Checkpoint 7
Decide whether the sequence
3, 8, 17, 30, 47, 68, . . .
can be represented perfectly by a linear or a quadratic model. If so, find the model.
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Appendix D
Example 8
■
Mathematical Induction
D9
Finding an Appropriate Model
Decide whether the sequence 2, 9, 16, 23, 30, 37, . . . can be represented perfectly by
a linear or a quadratic model. If so, find the model.
SOLUTION
Find the first differences of the sequence.
n:
1
2
3
4
5
6
an :
2
9
16
23
30
37
First differences:
7
7
7
7
7
Because, the first differences are all the same, the sequence can be represented perfectly
by a linear model of the form
an ⫽ an ⫹ b.
By substituting 1 and 2 for n, you can obtain a system of two linear equations in two
variables.
a1 ⫽ a共1兲 ⫹ b ⫽ 2
a2 ⫽ a共2兲 ⫹ b ⫽ 9
Substitute 1 for n.
Substitute 2 for n.
You now have a system of two equations in a and b.
冦2aa ⫹⫹ bb ⫽⫽ 29
Equation 1
Equation 2
Using the techniques discussed in Chapter 5, you can find that the solution of this
system is a ⫽ 7 and b ⫽ ⫺5. So, the linear model is
an ⫽ 7n ⫺ 5.
Checkpoint 8
Decide whether the sequence 6, 10, 14, 18, 22, 26, . . . can be represented
perfectly by a linear or a quadratic model. If so, find the model.
SUMMARIZE
1. State the principle of mathematical induction (page D2). For examples
of using the principle of mathematical induction to prove a formula, see
Examples 2 and 3.
2. Make a list of the formulas for the sums of powers of integers (page D5).
For an example that uses a formula to find a sum of powers of integers, see
Example 4.
3. Describe the guidelines for finding a formula for the nth term of a sequence
(page D6). For an example of finding a formula for the nth term of a
sequence, see Example 6.
4. Describe how to use the finite differences of a sequence to determine
whether the sequence can be represented perfectly by a linear or a quadratic
model (page D8). For examples of using finite differences to find models,
see Examples 7 and 8.
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Mathematical Induction
The following warm-up exercises involve skills that were covered in earlier sections. You will use
these skills in the exercise set for this section.
SKILLS WARM UP D
In Exercises 1– 4, find the sum.
6
兺
1.
5
共2k ⫺ 3兲
2.
k⫽3
兺
共 j2 ⫺ j兲
j⫽1
5
3.
兺 冢1 ⫹ i 冣
2
1
k⫽2 k
兺
4.
1
i⫽1
In Exercises 5–10, simplify the expression.
2共k ⫹ 1兲 ⫹ 3
5
5.
32k
8.
32共k⫹1兲
6.
3共k ⫹ 1兲 ⫺ 2
6
9.
k⫹1
k2 ⫹ k
7. 2
10.
⭈ 22共k⫹1兲
冪32
冪50
Exercises D
Using Pk to find Pk ⴙ1 In Exercises 1–4, find Pk⫹1 for
the given Pk . See Example 1.
5
1. Pk ⫽
k共k ⫹ 1兲
3. Pk ⫽
共k ⫹ 2兲共k ⫹ 3兲
k2
4. Pk ⫽
2共k ⫹ 1兲2
2
Using Mathematical Induction In Exercises 5–18,
use mathematical induction to prove the formula for
every positive integer n. See Examples 2 and 3.
5. 2 ⫹ 4 ⫹ 6 ⫹ 8 ⫹ . . . ⫹ 2n ⫽ n共n ⫹ 1兲
6. 3 ⫹ 7 ⫹ 11 ⫹ 15 ⫹ . . . ⫹ 共4n ⫺ 1兲 ⫽ n共2n ⫹ 1兲
n
7. 2 ⫹ 7 ⫹ 12 ⫹ 17 ⫹ . . . ⫹ 共5n ⫺ 3兲 ⫽ 共5n ⫺ 1兲
2
n
8. 1 ⫹ 4 ⫹ 7 ⫹ 10 ⫹ . . . ⫹ 共3n ⫺ 2兲 ⫽ 共3n ⫺ 1兲
2
2
3
n⫺1
n
⫽2 ⫺1
9. 1 ⫹ 2 ⫹ 2 ⫹ 2 ⫹ . . . ⫹ 2
2
3
n⫺1
.
.
.
⫹ 3 兲 ⫽ 3n ⫺ 1
10. 2共1 ⫹ 3 ⫹ 3 ⫹ 3 ⫹
n共n ⫹ 1兲
11. 1 ⫹ 2 ⫹ 3 ⫹ 4 ⫹ . . . ⫹ n ⫽
2
n2共n ⫹ 1兲2
12. 13 ⫹ 23 ⫹ 33 ⫹ 43 ⫹ . . . ⫹ n3 ⫽
4
13.
1
1
1
n
⫹
⫹. . .⫹
⫽
1共1 ⫹ 1兲 2共2 ⫹ 1兲
n共n ⫹ 1兲 n ⫹ 1
14.
冢
n
15.
兺
i⫽1
n
16.
兺
i⫽1
n
17.
兺
i⫽1
i5
冣冢
1
1⫹
2
冣冢
1
1⫹
3
冣...冢
冣
1
1⫹
⫽n⫹1
n
n2共n ⫹ 1兲2共2n2 ⫹ 2n ⫺ 1兲
⫽
12
i4 ⫽
n共n ⫹ 1兲共2n ⫹ 1兲共3n2 ⫹ 3n ⫺ 1兲
30
i共i ⫹ 1兲 ⫽
n共n ⫹ 1兲共n ⫹ 2兲
3
n
i⫽1
Finding a Sum of Powers of Integers In Exercises
19–28, find the sum using the formulas for the sums of
powers of integers. See Example 4.
20
19.
1
1⫹
1
1
兺 共2i ⫺ 1兲共2i ⫹ 1兲 ⫽ 2n ⫹ 1
3
2. Pk ⫽
共k ⫺ 1兲
4
k2
n
18.
50
兺n
20.
n⫽1
n⫽1
6
21.
10
兺n
2
22.
n⫽1
4
6
兺
24.
6
兺
兺n
5
n⫽1
共n2 ⫺ n兲
10
26.
n⫽1
27.
3
8
兺n
n⫽1
25.
兺n
n⫽1
5
23.
兺n
兺 共n
3
⫺ n2兲
n⫽1
共6i ⫺ 8i3兲
i⫽1
5
28.
兺 共2j
3
⫺ j 2兲
j⫽1
Checking Solutions In Exercises 29–34, determine
whether the inequality is true for n ⫽ 1, 2, 3, and 4.
29. n2 < n3
31. 共n ⫹ 1兲2 ⱕ n3
n⫹1
33. n ⫺ 1 ⱖ
n2
30. 4n4 ⱕ n6
32. 共n ⫺ 2兲3 < n2
共n ⫺ 1兲2
< n! ⫺ n
34.
n
Proving an Inequality by Mathematical Induction In
Exercises 35–38, prove the inequality for the indicated
integer values of n. See Example 5.
4
35. n! > 2n, n ⱖ 4
36. 共3 兲 > n, n ⱖ 7
1
1
1
1
⫹
⫹
⫹. . .⫹
> 冪n, n ⱖ 2
37.
冪1
冪2
冪3
冪n
x n⫹1
x n
<
, n ⱖ 1 and 0 < x < y
38.
y
y
n
冢冣
冢冣
9781133108490_App_D.qxp
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Page D11
Appendix D
Finding a Formula for a Finite Sum In Exercises
39–44, find a formula for the sum of the first n terms of
the sequence. See Example 6.
39. 1, 5, 9, 13, . . .
40. 25, 22, 19, 16, . . .
9 81 729
81
41. 1, 10, 100, 1000, . . .
42. 3, ⫺ 92, 27
4,⫺8,. . .
1 1 1 1
1
43. , , , , . . . ,
,. . .
4 12 24 40
2n共n ⫹ 1兲
44.
1
1
1
1
1
,
,
,
,. . .,
,. . .
2⭈3 3⭈4 4⭈5 5⭈6
共n ⫹ 1兲共n ⫹ 2兲
Proving Properties In Exercises 45–52, use
mathematical induction to prove the given property for
all positive integers n.
45. 共ab兲n ⫽ anbn
46.
冢冣
a
b
n
⫽
an
bn
47. If x1 ⫽ 0, x2 ⫽ 0, . . . , xn ⫽ 0, then
⫺1 ⫺1 . . . x⫺1.
共x1 x2 x3 . . . x n 兲⫺1 ⫽ x⫺1
1 x2 x3
n
48. If x1 > 0, x2 > 0, . . . , xn > 0, then
ln共x1 x2 x3 . . . xn 兲 ⫽
ln x1 ⫹ ln x2 ⫹ ln x3 ⫹ . . . ⫹ ln xn.
49. Generalized Distributive Law:
x共 y1 ⫹ y2 ⫹ . . . ⫹ yn 兲 ⫽ xy1 ⫹ xy2 ⫹ . . . ⫹ xyn
50. 共a ⫹ bi兲n and 共a ⫺ bi兲n are complex conjugates for all
n ⱖ 1.
51. A factor of 共n3 ⫹ 3n2 ⫹ 2n兲 is 3.
52. A factor of 共22n⫺1 ⫹ 32n⫺1兲 is 5.
53. Writing In your own words, explain what is meant by
a proof by mathematical induction.
54. Think About It What conclusion can be drawn from
the given information about the sequence of statements
Pn?
(a) P3 is true and Pk implies Pk⫹1.
(b) P1, P2, P3, . . . , P50 are all true.
(c) P1, P2, and P3 are all true, but the truth of Pk does
not imply the truth of Pk⫹1.
(d) P2 is true and P2k implies P2k⫹2.
Finding an Appropriate Model In Exercises 55–60,
decide whether the sequence can be represented
perfectly by a linear or a quadratic model. If so, find the
model. See Examples 7 and 8.
55.
56.
57.
58.
59.
60.
5, 13, 21, 29, 37, 45, . . .
2, 9, 16, 23, 30, 37, . . .
6, 15, 30, 51, 78, 111, . . .
0, 6, 16, 30, 48, 70, . . .
⫺2, 1, 6, 13, 22, 33, . . .
⫺1, 8, 23, 44, 71, 104, . . .
■
D11
Mathematical Induction
Using Finite Differences In Exercises 61–70, write
the first five terms of the sequence, where a1 ⫽ f 共1兲.
Then calculate the first and second differences of the
sequence. Does the sequence have either a linear model
or a quadratic model? If so, find the model.
61. f 共1兲 ⫽ 0
an ⫽ an⫺1
63. f 共1兲 ⫽ 3
an ⫽ an⫺1
65. a0 ⫽ 0
an ⫽ an⫺1
67. f 共1兲 ⫽ 2
an ⫽ an⫺1
69. a0 ⫽ 1
an ⫽ an⫺1
62. f 共1兲 ⫽ 2
an ⫽ n ⫺ an⫺1
64. f 共1兲 ⫽ ⫺3
an ⫽ ⫺2an⫺1
66. a0 ⫽ 2
an ⫽ 共an⫺1兲2
68. f 共1兲 ⫽ 0
an ⫽ an⫺1 ⫹ 2n
70. a0 ⫽ 0
an ⫽ an⫺1 ⫺ 1
⫹3
⫺n
⫹n
⫹2
⫹ n2
Finding a Quadratic Model In Exercises 71–74, find a
quadratic model for the sequence with the indicated
terms.
71.
72.
73.
74.
a0
a0
a1
a0
⫽ 3, a1 ⫽ 3, a4 ⫽ 15
⫽ 7, a1 ⫽ 6, a3 ⫽ 10
⫽ 0, a2 ⫽ 8, a4 ⫽ 30
⫽ ⫺3, a2 ⫽ ⫺5, a6 ⫽ ⫺57
75. Think About It When the second differences of a
sequence are all zero, can the sequence be represented
perfectly by a linear model? Explain.
76.
HOW DO YOU SEE IT? The table shows the
distance ds (in feet) required to stop a truck from
the moment the brakes are applied at a speed of
s miles-per-hour on dry, level pavement. The first
and second differences of the sequence of terms
ds are shown below the table. (Source: Virginia
Transportation Research Council)
Speed, s
10
20
30
40
50
60
70
Distance, ds
6
25
57
102
159
229
311
First differences: 19
Second differences:
32
13
45
13
57
12
70
13
82
12
(a) Will a linear or a quadratic model fit the data
best? Explain.
(b) When you drive a truck, do you assume an equal
risk for each 10 mile-per-hour increase in speed?
Explain your reasoning.