Copyright © 2007 Pearson Education, Inc. Slide 8-1.


Chapter 8: Further Topics in Algebra

8.1 Sequences and Series

8.2 Arithmetic Sequences and Series

8.3 Geometric Sequences and Series

8.4 The Binomial Theorem

8.5 Mathematical Induction

8.6 Counting Theory

8.7 Probability



• Mathematical induction is used to prove

statements claimed true for every positive

integer n.

For example, the summation rule

is true for each integer n > 1.

( 1)1 2 3 ...

2

n nn



Label the statement Sn.

For any one value of n, the statement can be

verified to be true.

( 1): 1 2 3 ...

2n

n nS n



To show Sn is true for every n requires mathematical induction.

1

2

3

1(1 1)1, : 1 true since 1=1

22(2 1)

2, : 1 2 true since 3=323(3 1)

3, : 1 2 3 true since 6=62

n S

n S

n S



Principle of Mathematical Induction

Let Sn be a statement concerning the positive integer n.

Suppose that

1. S1 is true;

2. For any positive integer k, k < n, if Sk is true,

then Sk+1 is also true.

Then, Sn is true for every positive integer n.



Proof by Mathematical Induction

Step 1 Prove that the statement is true for n = 1.

Step 2 Show that for any positive integer k, if Sk is

true, then Sk+1 is also true.


8.5 Proving an Equality Statement

Example Let Sn be the statement

Prove that Sn is true for every positive integer n.

Solution The proof uses mathematical induction.

( 1): 1 2 3 ...

2n

n nS n



Solution Step 1 Show that the statement is true when

n = 1. S1 is the statement

which is true since both sides equal 1.

1(1 1)1

2



Solution Step 2 Show that if Sk is true then Sk+1 is

also true. Start with Sk

and assume it is a true statement. Add k + 1 to

each side

( 1)1 2 3 ...

2

k kk

( 1)1 2 3 ... ( 1) ( 1)

2

k kk k k



Solution Step 2

( 1)1 2 3 ... ( 1) ( 1)

2

( 1) 12

2( 1)

2

k kk k k

kk

kk



Solution Step 2

This is the statement for n = k + 1. It has been shown

that if Sk is true then Sk+1 is also true. By

mathematical induction Sn is true for all positive integers n.

21 2 3 ... ( 1) ( 1)

2

( 1) 1( 1)

2

kk k k

kk



Generalized Principle of Mathematical Induction

Let Sn be a statement concerning the positive integer n.

Suppose that

Step 1 Sj is true;

Step 2 For any positive integer k, k > j, if Sk implies

Sk+1.

Then, Sn is true for all positive integers n > j.


8.5 Using the Generalized Principle

Example Let Sn represent the statement

Show that Sn is true for all values of n > 3.

Solution Since the statement is claimed to be true for

values of n beginning with 3 and not 1, the proof uses

the generalized principle of mathematical induction.

: 2 2 1nnS n



Solution Step 1 Show that Sn is true when n = 3. S3

is the statement

which is true since 8 > 7.

32 2 3 1



Solution Step 2 Show that Sk implies Sk+1 for k > 3.

Assume Sk

is true. Multiply each side by 2, giving

or

or, equivalently

2 2 1k k

2 2 2(2 1)k k

12 4 2k k

12 2( 1) 2 .k k k



Solution Step 2 Since k > 3, then 2k >1 and it follows

that

or

which is the statement Sk+1. Thus Sk implies Sk+1 and,

by the generalized principle, Sn is true for all n > 3.

12 2( 1) 2 2( 1) 1k k k k

12 2( 1) 1k k

Copyright © 2007 Pearson Education, Inc. Slide 8-1.

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