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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 My Solution | Comment 13 of 17 |
For all integers n,
  (n + 1) = n + 2n + 1 = (n) + (2n + 1)

For n = 1,
  (1 + 1) = (1) +2(1) + 1 = 1 + 3
For n = 2,
  (2 + 1) = (2) +2(2) + 1 = 4 + 5
For n = 3,
  (3 + 1) = (3) +2(3) + 1 = 9 + 7

For all integers n,
  (n + 1) = n + 2n + 1.
  → (n + 1) - n = 2n + 1

For n = 1,
  (2) - (1) = 2(1) +1 = 3
For n = 2,
  (3) - (2) = 2(2) +1 = 5
For n = 3,
  (4) - (3) = 2(3) +1 = 7

I forget what this method of proof is called or whether I really finished the proof, but I know I'm right. =]

  Posted by Charley on 2005-05-14 09:37:07
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