   Chapter 2.2, Problem 2E

Chapter
Section
Textbook Problem

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

To determine

To prove: That 1+3+5++(2n1)=n2 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

1+3+5++(2n1)=n2

Here, (2n1) 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]=21=1

And the value of right side is

(1)2=1.

Thus, P1 is true.

Assume now that Pk is true. That is, assume that the equation 1+3+5++(2k1)=k2 is true.

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

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