# To prove: 1 + 3 + 5 + ⋅ ⋅ ⋅ + ( 2 n − 1 ) = n 2 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1, Problem 18P
To determine

## To prove: 1+3+5+⋅⋅⋅+(2n−1)=n2.

Expert Solution

### Explanation of Solution

Proof:

Use Mathematical induction on n.

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

Base case: For n=1.

To prove that the statement is true for n=1.

(21)=12

Therefore, the statement is true for n=1.

Induction hypothesis: n=k

Assuming that the claim is true for n=k.

That is, sk=1+3+5++(2k1)=k2.

Inductive step: n=k+1

To prove the statement is true for n=k+1.

sk+1=1+3+5++(2k1)+(2k+1)=sk+(2k+1)=k2+(2k+1)=(k+1)2

Therefore, the statement is true for k+1.

Thus, 1+3+5++(2n1)=n2 is true for all positive integer n.

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