TITLE: Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
QUESTION [39 upvotes]: Prove that for all integers $n$, $n \geq 1$,
  $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$

How would I go about proving this?

REPLY [10 votes]: Proof that $n^2=1+3+5+\ldots+(2n-1)$ by subdividing the square of side $2n+1$ in two different ways: