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 k.

Inductive step: We wish to prove the claim for k+1.

Proof: 

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

Therefore, because we could have picked any value integer value of k for k>=1,
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 hypothesis: Assume the claim is true for k.  

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

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