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Odd Sum (Posted on 2002-08-08) Difficulty: 2 of 5
Prove that the sum of consecutive odd numbers beginning at 1 (eg 1, 3, 5, ..) always adds up to a perfect square

See The Solution Submitted by Cheradenine    
Rating: 3.9000 (10 votes)

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Solution geometric (not rigorous) solution/proof | Comment 12 of 17 |
all squares of integer size have perfect sq areas. A=s^2

starting from 1x1 square,

+

you can add 1 block horiz and 1 block vert to form this shape:

++
+

then you can add 1 corner block to complete a larger square sized 2x2

++
++

For square of size N, you can add 2N blocks(one horiz, one vertical) and 1 corner block to complete a larger square of size (N+1)

+++ ... +
+++ ... +
.
.
+++ ... +

thus starting from N=1, Square of 1x1, area=1, you can add odd number 2N+1 of blocks to form another square, where its area is a perfect square. 2N+1 is conseq sequence of numbers 3,5,7 as N increases by 1 every repetiton.

As stated by others N^2+2N+1 = (N+1)^2, but it's easier to visualize the squares.
  Posted by karrio on 2004-06-08 12:06:39
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