   Chapter 2.2, Problem 4E

Chapter
Section
Textbook Problem

Prove that the statements in Exercises 1 − 16 are true for every positive integer n . 1 2 + 3 2 + 5 2 + ⋅ ⋅ ⋅ + ( 2 n − 1 ) 2 = n ( 2 n − 1 ) ( 2 n + 1 ) 3

To determine

To prove: That 12+32+52++(2n1)2=n(2n1)(2n+1)3 is true for all n by using mathematical induction.

Explanation

Formula Used:

Considering the given statement is Pn for all integers n,

a. if Pn is true for n=1

b. if the truth of Pk always implies that Pk+1 is true, then the statement Pn is true for all positive integers n.

Proof:

For each positive integer n, let Pn be the statement

12+32+52++(2n1)2=n(2n1)(2n+1)3

Here, (2n1)2 is the last term on the left side.

When n=1, there is only one term, and no addition is actually performed.

When n=1, the value of the left side is

[2(1)1]2=(1)2=1

And the value of right side is

(1)[2(1)1][2(1)+1]3=1133=1

Thus P1 is true.

Assume now that Pk is true. That is, assume that the equation 12+32+52++(2k1)2=k(2k1)(2k+1)3 is true.

With this assumption made, prove that Pk+1 is true. Add

[2(k+1)1]2=(2k+1)2 to both sides of the assumed equality, the following result is obtained

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