Prove the following by induction1+3+5+...+(2n-1)=n*n

1.	Prove the following by induction1+3+5+...+(2n-1)=n*n
Answer» Proof by induction on n: Step 1: prove that the equation is valid when n = 1 When n = 1, we have (2(1) - 1) = 12, so the statement holds for n = 1. Step 2: Assume that the equation is true for n, and prove that the equation is true for n + 1. Assume: 1 + 3 + 5 + ... + (2n - 1) = n2 Prove: 1 + 3 + 5 +...+ (2(n + 1) - 1) = (n + 1)2 Proof: 1 + 3 + 5 +... + (2(n + 1) - 1) = 1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1) = n2+ (2n + 2 - 1) (by assumption) = n2+ 2n + 1 = (n + 1)2 So, by induction, for every positive integer n, 1 + 3 + 5 + ... + (2n - 1) = n2.

Answer»

Proof by induction on n:

Step 1: prove that the equation is valid when n = 1 When n = 1, we have (2(1) - 1) = 12, so the statement holds for n = 1.

Step 2: Assume that the equation is true for n, and prove that the equation is true for n + 1.

Assume: 1 + 3 + 5 + ... + (2n - 1) = n2

Prove: 1 + 3 + 5 +...+ (2(n + 1) - 1) = (n + 1)2 Proof: 1 + 3 + 5 +... + (2(n + 1) - 1)

= 1 + 3 + 5 + ... + (2n - 1) + (2n + 2 - 1)

= n2+ (2n + 2 - 1) (by assumption)

= n2+ 2n + 1

= (n + 1)2

So, by induction, for every positive integer n,

1 + 3 + 5 + ... + (2n - 1) = n2.

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