Simple induction Proof
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Claim: The sum of the first n odd numbers is equal to n^2

Base case (n=1): the sum of the 1 odd numbers is just 1 = 1^2

Inductive Hypothesis: Assume the claim is true for n.

Inductive step: We wish to prove the claim for n+1: From the induction
hypothesis, we know that the sum of the first n numbers is n^2. We also
know the (n+1)th odd number is equal to 2n+1.  So the sum of the first n+1
odd numbers is n^2 + 2n + 1 = (n+1)^2.  Thus, proving the claim for n+1.

Therefore, by induction, the claim is true for all n >= 1.

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A more mathematical way
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Claim: From i = 1 to n, sum(2i-1) = n^2

Base case (n=1): sum(1) = 1 = 1^2

Inductive step: Assume the claim is true for n.  We wish to prove the
claim for n+1: From i=1 to n+1, sum(2i-1) = From i=1 to n, sum(2i-1) +
2(n+1)-1
			   = n^2 + 2(n+1) - 1    <--Induction hypothesis
			   = n^2 + 2n + 1
			   = (n+1)^2

Therefore, by induction, the claim is true for all n >= 1.