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

 Submitted by Cheradenine Rating: 3.9000 (10 votes) Solution: (Hide) Consider the sum of n odd numbers We can pair up symmetrical elements that add to a certain constant and then multiply by the number of pairs (ie Gauss) The largest odd number is 2n - 1. If n is even, the sum S: S = each pair * number of pairs S = (2n-1 + 1) * (n / 2) S = 2n * n/2 = nē If n is odd, the numbers do not pair out exactly, there is a remaining central term. However note that this term occurs in the middle so it is equal to 2n/2 = n. S = (each pair * number of pairs) + center S = 2n * (n-1/2) + n (remember n is odd) S = 2n * (n-1) + n S = nē - n + n = nē For any n, the sum is always a perfect square. (See the comments for other interesting ways to prove this)

 Subject Author Date re: What, me worry? broll 2015-04-06 03:56:52 What, me worry? Steve Herman 2014-11-17 21:28:48 Complete solution danish ahmed khan 2012-10-23 14:38:05 Answer K Sengupta 2007-03-24 12:00:10 My Solution Charley 2005-05-14 09:37:07 geometric (not rigorous) solution/proof karrio 2004-06-08 12:06:39 nevermind elson 2004-05-08 06:26:59 I know this is late but... elson 2004-05-08 06:25:29 re(2): Difference Method NO SHIFT Richard 2004-04-05 14:18:25 re: Difference Method SHIFT BY ONE Ady TZIDON 2004-04-05 10:27:50 Difference Method Richard 2004-04-04 18:12:59 Simplify a bit Jack Squat 2004-01-05 14:17:06 re(2): (Happy's post) TomM 2002-08-08 21:53:20 re: (Happy's post) TomM 2002-08-08 21:21:28 re: Arithmetic series lucky 2002-08-08 10:32:46 Arithmetic series lucky 2002-08-08 09:57:06 No Subject Happy 2002-08-08 08:42:19

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