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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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re(2): (Happy's post) | Comment 5 of 17 |
(In reply to re: (Happy's post) by TomM)

Alternatively you could start at exactly the point at whic Happy left off:

Adding the next odd number of counters to the square array is eqivalent to adding another row and another colomn.

Given an N x N square array (of N² counters) adding a new row means adding N counters and acheiving an (N = !) x N array of N² + N = N(N + 1) counters. Adding a new column means addind another (N + 1) counters and acheiving an (N + 1) x (N + 1) array of (N + 1)² = N² + 2N +1 counters. The total number of counters added is 2N + 1, which is an odd number. It is easy to show that the N x N square can be considered to be a (N - 1) x (N - 1) square with an added row and column totalling 2N - 1 counters, so the added numbers are consecutive odd numbers.

Since Happy showed the first few examples to work out imperically, this completes the inductive proof.
  Posted by TomM on 2002-08-08 21:53:20

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