Prove that the sum of consecutive odd numbers beginning at 1 (eg 1, 3, 5, ..) always adds up to a perfect square

(In reply to re: Difference Method SHIFT BY ONE by Ady TZIDON)

Ady:

You are mistaken. I wrote

"1 + 3 + ... + 2N+1 = (1-0) + (4-1) + (9-4) + ... + ((N+1)^2 - N^2)"

Notice that 2N+1 is (N+1)^2 - N^2.

Filling in also the next-to-last term gives

1 + 3 + ... + 2N+1 = (1-0) + (4-1) + (9-4) + ... +( N^2 - (N-1)^2) + ((N+1)^2 - N^2)

and it is thus clear that the only term that does not cancel out is (N+1)^2 just as I wrote. Also, consider 1 + 3 = 1 + (2*1+1) --  here N=1, (N+1)^2=2^2=4=1+3.

Posted by Richard on 2004-04-05 14:18:25