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C. The General Principle of Mathematical Induction

# C. The General Principle of Mathematical Induction

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CHAPTER 9 Additional Topics in Algebra

2kϩ1 ϭ 212k 2

Ն 21k ؉ 12

Ն 2k ϩ 2

properties of exponents

induction hypothesis: substitute k ϩ 1 for 2k

(symbol changes since k ϩ 1 is less than or equal to 2k)

distribute

Since k is a positive integer, 2kϩ1 Ն 2k ϩ 2 Ն k ϩ 2,

showing 2kϩ1 Ն k ϩ 2.

WORTHY OF NOTE

Note there is no reference to an, ak,

or ak+1 in the statement of the

general principle of mathematical

induction.

EXAMPLE 4

Since the truth of Pkϩ1 follows from Pk, the formula is true for all n.

Now try Exercises 39 through 42

Proving Divisibility Using Mathematical Induction

Let Pn be the statement, “4n Ϫ 1 is divisible by 3 for all positive integers n.” Use

mathematical induction to prove that Pn is true.

Solution

If a number is evenly divisible by three, it can be written as the product of 3 and

some positive integer we will call p.

1. Show Pn is true for n ϭ 1:

Pn: 4n Ϫ 1 ϭ 3p

P1: 4112 Ϫ 1 ϭ 3p

3 ϭ 3p ✓

given statement, p ʦ ‫ޚ‬

substitute 1 for n

statement is true for n ϭ 1

2. Assume that Pk is true.

Pk:

4k Ϫ 1 ϭ 3p

4k ϭ 3p ϩ 1

induction hypothesis

isolate 4k

and use it to show the truth of Pkϩ1. That is,

Pkϩ1:

4kϩ1 Ϫ 1 ϭ 3q for q ʦ ‫ ޚ‬is also true.

Beginning with the left-hand side we have:

4kϩ1 Ϫ 1 ϭ 4 # 4k Ϫ 1

ϭ 4 # 13p ؉ 12 Ϫ 1

ϭ 12p ϩ 3

ϭ 314p ϩ 12 ϭ 3q

properties of exponents

induction hypothesis: substitute 3p ϩ 1 for 4k

distribute and simplify

factor

The last step shows 4

Ϫ 1 is divisible by 3. Since the original statement is

true for n ϭ 1, and the truth of Pk implies the truth of Pkϩ1, the statement,

“4n Ϫ 1 is divisible by 3” is true for all positive integers n.

kϩ1

Now try Exercises 43 through 47

C. You’ve just seen how

we can apply the principle of

mathematical induction to

general statements involving

natural numbers

We close this section with some final notes. Although the base step of a proof by

induction seems trivial, both the base step and the induction hypothesis are necessary

1

1

parts of the proof. For example, the statement n 6

is false for n ϭ 1, but true for

3

3n

all other positive integers. Finally, for a fixed natural number p, some statements are

false for all n 6 p, but true for all n Ն p. By modifying the base case to begin at p, we

can use the induction hypothesis to prove the statement is true for all n greater than p.

For example, n 6 13n2 is false for n 6 4, but true for all n Ն 4.

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Section 9.4 Mathematical Induction

9.4 EXERCISES

CONCEPTS AND VOCABULARY

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

1. No

statement

number of verifications can prove a

true.

3. Assuming that a statement/formula is true for

n ϭ k is called the

.

5. Explain the equation Sk ϩ akϩ1 ϭ Skϩ1. Begin by

saying, “Since the kth term is arbitrary p ” (continue

from here).

7. an ϭ 10n Ϫ 6

9. an ϭ n

11. an ϭ 2nϪ1

8. an ϭ 6n Ϫ 4

10. an ϭ 7n

12. an ϭ 213nϪ1 2

For the given sum formula Sn, find S4, S5, Sk, and Sk؉1.

13. Sn ϭ n15n Ϫ 12

15. Sn ϭ

n1n ϩ 12

2

17. Sn ϭ 2n Ϫ 1

14. Sn ϭ n13n Ϫ 12

16. Sn ϭ

7n1n ϩ 12

2

18. Sn ϭ 3n Ϫ 1

Verify that S4 ؉ a5 ‫ ؍‬S5 for each exercise. Note that

each Sn is identical to those in Exercises 13 through 18.

19. an ϭ 10n Ϫ 6; Sn ϭ n15n Ϫ 12

20. an ϭ 6n Ϫ 4; Sn ϭ n13n Ϫ 12

21. an ϭ n; Sn ϭ

n1n ϩ 12

22. an ϭ 7n; Sn ϭ

23. an ϭ 2

nϪ1

2

7n1n ϩ 12

2

; Sn ϭ 2 Ϫ 1

n

24. an ϭ 213nϪ1 2; Sn ϭ 3n Ϫ 1

WORKING WITH FORMULAS

25. Sum of the first n cubes (alternative form):

(1 ؉ 2 ؉ 3 ؉ 4 ؉ p ؉ n)2

Earlier we noted the formula for the sum of the

n2 1n ϩ 12 2

. An alternative is

first n cubes was

4

given by the formula shown.

a. Verify the formula for n ϭ 1, 5, and 9.

4. The graph of a sequence is

, meaning it

is made up of distinct points.

6. Discuss the similarities and differences between

mathematical induction applied to sums and the

general principle of mathematical induction.

For the given nth term an, find a4, a5, ak, and ak؉1.

2. Showing a statement is true for n ϭ 1 is called the

of an inductive proof.

b. Verify the formula using

1ϩ2ϩ3ϩpϩnϭ

n1n ϩ 12

2

.

26. Powers of the imaginary unit: in ؉ 4 ‫ ؍‬in, where

i ‫ ؍‬1 ؊1

Use a proof by induction to prove that powers of

the imaginary unit are cyclic. That is, that they

cycle through the numbers i, Ϫ1, Ϫi, and 1 for

consecutive powers.

APPLICATIONS

Use mathematical induction to prove the indicated sum

formula is true for all natural numbers n.

27. 2 ϩ 4 ϩ 6 ϩ 8 ϩ 10 ϩ p ϩ 2n;

an ϭ 2n, Sn ϭ n1n ϩ 12

28. 3 ϩ 7 ϩ 11 ϩ 15 ϩ 19 ϩ p ϩ 14n Ϫ 12;

an ϭ 4n Ϫ 1, Sn ϭ n12n ϩ 12

29. 5 ϩ 10 ϩ 15 ϩ 20 ϩ 25 ϩ p ϩ 5n;

5n1n ϩ 12

an ϭ 5n, Sn ϭ

2

30. 1 ϩ 4 ϩ 7 ϩ 10 ϩ 13 ϩ p ϩ 13n Ϫ 22;

n13n Ϫ 12

an ϭ 3n Ϫ 2, Sn ϭ

2

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31. 5 ϩ 9 ϩ 13 ϩ 17 ϩ p ϩ 14n ϩ 12;

an ϭ 4n ϩ 1, Sn ϭ n12n ϩ 32

32. 4 ϩ 12 ϩ 20 ϩ 28 ϩ 36 ϩ p ϩ 18n Ϫ 42;

an ϭ 8n Ϫ 4, Sn ϭ 4n2

37.

1

1

1

1

ϩ

ϩ

ϩpϩ

;

1132

3152

5172

12n Ϫ 1212n ϩ 12

1

n

an ϭ

, Sn ϭ

2n ϩ 1

12n Ϫ 12 12n ϩ 12

1

1

1

1

ϩ

ϩ

ϩpϩ

;

1122

2132

3142

n1n ϩ 12

1

n

, Sn ϭ

an ϭ

n

ϩ

1

n1n ϩ 12

Use the principle of mathematical induction to prove

that each statement is true for all natural numbers n.

33. 3 ϩ 9 ϩ 27 ϩ 81 ϩ 243 ϩ p ϩ 3n;

313n Ϫ 12

an ϭ 3n, Sn ϭ

2

34. 5 ϩ 25 ϩ 125 ϩ 625 ϩ p ϩ 5n;

515n Ϫ 12

an ϭ 5n, Sn ϭ

4

35. 2 ϩ 4 ϩ 8 ϩ 16 ϩ 32 ϩ 64 ϩ p ϩ 2n;

an ϭ 2n, Sn ϭ 2nϩ1 Ϫ 2

38.

39. 3n Ն 2n ϩ 1

40. 2n Ն n ϩ 1

41. 3 # 4nϪ1 Յ 4n Ϫ 1

42. 4 # 5nϪ1 Յ 5n Ϫ 1

43. n2 Ϫ 7n is divisible by 2

36. 1 ϩ 8 ϩ 27 ϩ 64 ϩ 125 ϩ 216 ϩ p ϩ n3;

n2 1n ϩ 12 2

an ϭ n3, Sn ϭ

4

44. n3 Ϫ n ϩ 3 is divisible by 3

45. n3 ϩ 3n2 ϩ 2n is divisible by 3

46. 5n Ϫ 1 is divisible by 4

47. 6n Ϫ 1 is divisible by 5

EXTENDING THE CONCEPT

48. You may have noticed that the sum formula for the first n integers was quadratic, and the formula for the first n

integer squares was cubic. Is the formula for the first n integer cubes, if it exists, a quartic (degree four) function?

Use your calculator to run a quartic regression on the first five perfect cubes (enter 1 through 5 in L1 and the

cumulative sums in L2). What did you find? How is this exercise related to Exercise 36?

xn Ϫ 1

ϭ 11 ϩ x ϩ x2 ϩ x3 ϩ p ϩ xnϪ1 2.

xϪ1

50. Use mathematical induction to prove that for 14 ϩ 24 ϩ 34 ϩ p ϩ n4, where an ϭ n4,

49. Use mathematical induction to prove that

Sn ϭ

n1n ϩ 12 12n ϩ 12 13n2 ϩ 3n Ϫ 12

30

.

51. (7.2) Given the matrices A ϭ c

Ϫ1

3

53. (1.1) State the equation of the circle whose graph

is shown here.

2

d and

1

2 Ϫ1

d , find A ϩ B, A Ϫ B, 2A Ϫ 3B,

4

3

AB, BA, and BϪ1.

Bϭ c

52. (2.5) State the domain and

range of the piecewise

function shown here.

y

10

(1, 7)

8

6

(4, 3)

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

y

5

4

3

(Ϫ1, 1) 2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3

(3, Ϫ2)

5 x

2 4 6 8 10 x

54. (5.5) Solve: 3e2xϪ1 ϩ 5 ϭ 17. Answer in exact

form.

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Reinforcing Basic Concepts

MID-CHAPTER CHECK

1. an ϭ 7n Ϫ 4

Find the number of terms in each series and then find

the sum. Verify results on a graphing calculator.

12. 2 ϩ 5 ϩ 8 ϩ 11 ϩ p ϩ 74

2. an ϭ n2 ϩ 3

13.

In Exercises 1 through 3, the nth term is given. Write

the first three terms of each sequence and find a9.

3. an ϭ 1Ϫ12 n 12n Ϫ 12

͚3

ϩ 32 ϩ 52 ϩ 72 ϩ p ϩ 31

2

14. For an arithmetic series, a3 ϭ Ϫ8 and a7 ϭ 4.

Find S10.

4

4. Evaluate the sum

1

2

nϩ1

15. For a geometric series, a3 ϭ Ϫ81 and a6 ϭ 3.

Find S10.

nϭ1

5. Rewrite using sigma notation.

1 ϩ 4 ϩ 7 ϩ 10 ϩ 13 ϩ 16

16. Identify a1 and the common ratio r. Then find an

expression for the general term an.

a. 2, 6, 18, 54, p

Match each formula to its correct description.

n1a1 ϩ an 2

6. Sn ϭ

7. an ϭ a1rnϪ1

2

a1

8. Sq ϭ

9. an ϭ a1 ϩ 1n Ϫ 12d

1Ϫr

a1 11 Ϫ rn 2

10. Sn ϭ

1Ϫr

a. sum of an infinite geometric series

1

b. 12, 14, 18, 16

,p

17. Find the number of terms in the series then compute

the sum. 541 ϩ 181 ϩ 16 ϩ p ϩ 812

18. Find the infinite sum (if it exists).

Ϫ49 ϩ 1Ϫ72 ϩ 1Ϫ12 ϩ 1Ϫ17 2 ϩ p

19. Barrels of toxic waste are stacked at a storage facility

in pyramid form, with 60 barrels in the first row, 59 in

the second row, and so on, until there are 10 barrels in

the top row. How many barrels are in the storage

facility? Verify results using a graphing calculator.

b. nth term formula for an arithmetic series

c. sum of a finite geometric series

d. summation formula for an arithmetic series

20. As part of a conditioning regimen, a drill sergeant

orders her platoon to do 25 continuous standing

broad jumps. The best of these recruits was able to

jump 96% of the distance from the previous jump,

with a first jump distance of 8 ft. Use a sequence/

series to determine the distance the recruit jumped

on the 15th try, and the total distance traveled by the

recruit after all 25 jumps. Verify results using a

graphing calculator.

e. nth term formula for a geometric series

11. Identify a1 and the common difference d. Then find

an expression for the general term an.

a. 2, 5, 8, 11, p

b. 32, 94, 3, 15

4,p

REINFORCING BASIC CONCEPTS

Applications of Summation

The properties of summation play a large role in the development of key ideas in a first semester calculus course,

and the following summation formulas are an integral part of these ideas. The first three formulas were verified in

Section 9.4, while proof of the fourth was part of Exercise 48 on page 802.

n

(1)

͚

n

c ϭ cn

iϭ1

(2)

͚

iϭ1

n1n ϩ 12

2

n

(3)

͚

i2 ϭ

n1n ϩ 1212n ϩ 12

iϭ1

6

n

(4)

͚

i3 ϭ

iϭ1

n2 1n ϩ 12 2

4

To see the various ways they can be applied consider the following.

Illustration 1 ᮣ Over several years, the owner of Morgan’s LawnCare has noticed that the company’s monthly

profits (in thousands) can be approximated by the sequence an ϭ 0.0625n3 Ϫ 1.25n2 ϩ 6n, with the points plotted in

Figure 9.52 (the continuous graph is shown for effect only). Find the company’s approximate annual profit.

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Solution ᮣ The most obvious approach would be to simply compute terms a1

through a12 (January through December) and find their sum: sum(seq(Y1, X,

1, 12), which gives a result of 35.75 or \$35,750.

As an alternative, we could add the amount of profit earned by the company

in the first 8 months, then add the amount the company lost (or broke even) during

the last 4 months. In other words, we could apply summation property IV:

12

͚a

n

8

ϭ

Figure 9.52

12

0

12

͚a

12

n

ϩ

͚a

n

[(see Figure 9.53), which gives the same result:

42 ϩ 1Ϫ6.252 ϭ 35.75 or \$35,750].

As a third option, we could use summation properties along with the

appropriate summation formulas, and compute the result manually. Note the

function is now written in terms of “i.” Distribute summation and factor out

constants (properties II and III):

iϭ1

iϭ1

12

͚

iϭ1

iϭ9

10.0625i3 Ϫ 1.25i2 ϩ 6i2 ϭ 0.0625

12

͚

12

i3 Ϫ 1.25

iϭ1

͚

Ϫ5

Figure 9.53

12

i2 ϩ 6

iϭ1

͚i

iϭ1

Replace each summation with the appropriate summation formula,

then substitute 12 for n:

ϭ 0.0625 c

ϭ 0.0625 c

n2 1n ϩ 12 2

4

2

1122 1132 2

d Ϫ 1.25 c

d Ϫ 1.25 c

n1n ϩ 12 12n ϩ 12

6

1122 1132 1252

4

6

ϭ 0.0625160842 Ϫ 1.2516502 ϩ 61782

ϭ 35.75

d ϩ 6c

d ϩ 6c

n1n ϩ 12

1122 1132

2

2

d

d

As we expected, the result shows profit was \$35,750. While some approaches seem “easier” than others, all have

great value, are applied in different ways at different times, and are necessary to adequately develop key concepts in

future classes.

Exercise 1: Repeat Illustration 1 if the profit sequence is an ϭ 0.125x3 Ϫ 2.5x2 ϩ 12x.

9.5

Counting Techniques

LEARNING OBJECTIVES

In Section 9.5 you will see

how we can:

A. Count possibilities using

B.

C.

D.

E.

lists and tree diagrams

Count possibilities using

the fundamental principle

of counting

Quick-count

distinguishable

permutations

Quick-count

nondistinguishable

permutations

Quick-count using

combinations

How long would it take to estimate the number of fans sitting shoulder-to-shoulder at

a sold-out basketball game? Well, it depends. You could actually begin counting 1, 2,

3, 4, 5, p , which would take a very long time, or you could try to simplify the process

by counting the number of fans in the first row and multiplying by the number of rows.

Techniques for “quick-counting” the objects in a set or various subsets of a large set

play an important role in a study of probability.

A. Counting by Listing and Tree Diagrams

Consider the simple spinner shown in Figure 9.54, which is divided into three equal parts. What are the different possible outcomes for two spins, spin 1 followed by spin 2? We might begin by

organizing the possibilities using a tree diagram. As the name implies, each choice or possibility appears as the branch of a tree,

with the total possibilities being equal to the number of (unique)

Figure 9.54

B

A

C

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Section 9.5 Counting Techniques

Figure 9.55

paths from the beginning point to the end of a

branch. Figure 9.55 shows how the spinner exercise would appear (possibilities for two spins).

Moving from top to bottom we can trace nine

possible paths: AA, AB, AC, BA, BB, BC, CA,

CB, and CC.

Begin

A

A

EXAMPLE 1

B

B

C

A

B

C

C

A

B

C

Listing Possibilities Using a Tree Diagram

A basketball player is fouled and awarded three free throws. Let H represent the

possibility of a hit (basket is made), and M the possibility of a miss. Determine the

possible outcomes for the three shots using a tree diagram.

Solution

Each shot has two possibilities, hit (H) or miss (M), so the tree will branch in two

directions at each level. As illustrated in the figure, there are a total of eight

possibilities: HHH, HHM, HMH, HMM, MHH, MHM, MMH, and MMM.

Begin

H

M

H

H

M

M

H

H

M

H

M

M

H

M

Now try Exercises 7 through 10

WORTHY OF NOTE

Sample spaces may vary

depending on how we define the

experiment, and for simplicity’s

sake we consider only those

experiments having outcomes that

are equally likely.

To assist our discussion, an experiment is any task that can be done repeatedly and has

a well-defined set of possible outcomes. Each repetition of the experiment is called a

trial. A sample outcome is any potential outcome of a trial, and a sample space is a

set of all possible outcomes.

In our first illustration, the experiment was spinning a spinner, there were three

sample outcomes (A, B, or C), the experiment had two trials (spin 1 and spin 2), and

there were nine elements in the sample space. Note that after the first trial, each of the

three sample outcomes will again have three possibilities (A, B, and C). For two trials

we have 32 ϭ 9 possibilities, while three trials would yield a sample space with

33 ϭ 27 possibilities. In general, for N equally likely outcomes we have

A “Quick-Counting” Formula for a Sample Space

If an experiment has N sample outcomes that are equally likely and the experiment

is repeated t times, the number of elements in the sample space is

N t.

EXAMPLE 2

Counting the Outcomes in a Sample Space

Many combination locks have the digits 0 through 39 arranged

along a circular dial. Opening the lock requires stopping at a

sequence of three numbers within this range, going

counterclockwise to the first number, clockwise to the second,

and counterclockwise to the third. How many three-number

combinations are possible?

5

35

10

30

25

15

20

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C. The General Principle of Mathematical Induction

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