Odd Number Theorem

Theorem: $$\sum_{i=1}^n (2n-1)=n^2$$

Proof by Induction
Base case: $$n=1$$. $$\sum_{i=1}^1 (2i-1)=(2\cdot1-1)=1=1^2$$ so the base case holds.

Inductive hypothesis: $$\sum_{i=1}^k (2k-1) = k^2\;\;\;\forall k>1$$

Inductive step: Consider $$n+1$$ $$\sum_{i=1}^{n+1}(2i-1) = \sum_{i=1}^n (2i-1) + [2(n+1)-1]$$ $$=n^2+[2(n+1)-1]$$ by the inductive hypothesis $$=n^2+2n+1$$ $$=(n+1)^2$$

So we are done by the Principle of Mathematical Induction.