(adsbygoogle = window.adsbygoogle || []).push({}); "Elegant" Proof of some summations

Ever since I was in the 8th grade sigma notation has been one of my favorite things to study and, more specifically, the things you can sum together. The proof was summing the numbers 1 to n is very simple and everyone knows the story about Gauss and that particular summation but I haven't seen a proof I like for sums of squares, cubics and quartics 1 to n (or any beyond that). I came up with the equations when I was a lot younger but I had no concept of a proof and I'm really stumped so, to get the ball rolling, can anyone point me to an "elegant" proof of the summation formula for squares 1^2 to n^2 or at least put me on the right wavelength of thought? The book I have gives a really dumb one that I don't like at all.

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# Elegant Proof of some summations

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