Summing Odd Numbers Geometrically

Date: 10/30/1999 at 04:16:39
From: Can Turkseven
Subject: Geometric proof

Can you prove that

     1 + 3 + 5 +... + (2n-1) = n*n

by using a simple geometric method?

Thanks.

Date: 10/30/1999 at 06:53:57
From: Doctor Floor
Subject: Re: Geometric proof

Hi Can,

Thanks for your question.

Let us consider a square formed out of smaller squares:

     -------------
     |  |  |  |  |
     -------------
     |  |  |  |  |
     -------------
     |  |  |  |  |
     -------------
     |  |  |  |  |
     -------------

We can build up such a square in layers. First we start with one 
square (marked with 1). Then we put a layer around this first square 
to complete a 2x2 square (marked with 2), and so on.

     -------------
     | 4| 4| 4| 4|
     -------------
     | 3| 3| 3| 4|
     -------------
     | 2| 2| 3| 4|
     -------------
     | 1| 2| 3| 4|
     -------------

The numbers of squares in these layers are 1, 3, 5, 7, ..., 2n-1.

And that proves that 1 + 3 + 5 + ... + (2n-1) = n*n.

I hope this helped.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/