Summing Odd Numbers GeometricallyDate: 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/ |
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