Proof Without Words II

Conjecture: For every natural number n, the sum of the first n odd numbers is n2.

In other words:

1 + 3 + 5 + . . . + (2n – 1) = n2

For example, starting with n = 1:

1 = 12

1 + 3 = 22

1 + 3 + 5 = 32

1 + 3 + 5 + 7 = 42

1 + 3 + 5 + 7 + 9 = 52

1 + 3 + 5 + 7 + 9 + 11 = 62

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 = 22. Similarly, the first three odd numbers (1,3, and 5) make a 3 x 3 square. Hence, 1 + 3 + 5 = 32. How cool is that?!

And there’s more: this elegant scheme is also a geometric depiction of the identity (k + 1)2 = k2 + 2k + 1. Can you spot this proof?

[Source: Excursions in Calculus]

***

Also check out an earlier post: Proof without Words.

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3 responses to “Proof Without Words II

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

  2. 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.

  3. @ 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!)

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