Geometric Proof

I was reminded of this equation while reading up on Andrey Kolomogorov (1903 - 1987) who spotted the relationship at the tender age of 5 (precosious brat). The relation was first mentioned by Pythagoras and his disciples but the first proof was elucidated later by Hipparchus.

We are seeking to prove

The sum of the first n odd numbers equals n².
We will begin by showing that the relationship holds true for first two odd numbers (n=2), i.e. 1 + 3.
Moving quickly on we will now show that the relationship holds true for the first 3 odd numbers (n=3), i.e. 1 + 3 + 5.
We can see clearly that each successive odd number added to the sequence will add an additional row and column to the rearranged image. Q.E.D.