**Conjecture:** For every natural number *n*, the sum of the first *n* odd numbers is *n*^{2}.

In other words:

1 + 3 + 5 + . . . + (2

n– 1) =n^{2}

For example, starting with *n* = 1:

1 = 1

^{2}1 + 3 = 2

^{2}1 + 3 + 5 = 3

^{2}1 + 3 + 5 + 7 = 4

^{2}1 + 3 + 5 + 7 + 9 = 5

^{2}1 + 3 + 5 + 7 + 9 + 11 = 6

^{2}And so on.

This conjecture can be proven by induction, but I think this little picture below does it marvelously and most succinctly:

If you start from the lower left corner, you can see that the first two odd numbers (1 and 3) make a 2 x 2 square. Hence, 1 + 3 = 2^{2}. Similarly, the first three odd numbers (1,3, and 5) make a 3 x 3 square. Hence, 1 + 3 + 5 = 3^{2}. How cool is that?!

And there’s more: this elegant scheme is also a geometric depiction of the identity (*k* + 1)^{2} = *k*^{2} + 2*k* + 1. Can you spot this proof?

[Source: *Excursions in Calculus*]

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Also check out an earlier post: *Proof without Words*.

It’s really cool! So simple, yet such a perfect proof!

Thanks for this one, Vishal! I’ve almost been disconnected from conventional/fundamental Mathematics and its aspects. Your awesome articles off and on help me reestablish the fading link.

@ Ganesh, @Rahul – Thanks for letting me know that I am not the only one who thought this visual proof is splendid! 🙂

@Rahul – I hear you. I think recreational math is the best way to reestablish that link. Most of my posts in the ‘Numbers’ category are my efforts to keep that link alive (and kicking!)