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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

  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)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: What, me worry?broll2015-04-06 03:56:52
SolutionWhat, me worry?Steve Herman2014-11-17 21:28:48
Complete solutiondanish ahmed khan2012-10-23 14:38:05
SolutionAnswerK Sengupta2007-03-24 12:00:10
SolutionMy SolutionCharley2005-05-14 09:37:07
Solutiongeometric (not rigorous) solution/proofkarrio2004-06-08 12:06:39
nevermindelson2004-05-08 06:26:59
I know this is late but...elson2004-05-08 06:25:29
re(2): Difference Method NO SHIFTRichard2004-04-05 14:18:25
Hints/Tipsre: Difference Method SHIFT BY ONEAdy TZIDON2004-04-05 10:27:50
Difference MethodRichard2004-04-04 18:12:59
Simplify a bitJack Squat2004-01-05 14:17:06
re(2): (Happy's post)TomM2002-08-08 21:53:20
re: (Happy's post)TomM2002-08-08 21:21:28
re: Arithmetic serieslucky2002-08-08 10:32:46
Some ThoughtsArithmetic serieslucky2002-08-08 09:57:06
Hints/TipsNo SubjectHappy2002-08-08 08:42:19
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