Three basic formulas for integer sums are given in every calculus textbook. See page 369 of Stewart's Calculus:Early Transcendentals (6'th Edition), for example. These formulas are useful for evaluating and approximating integrals using Riemann sums, among other things.
The inquisitive student may wonder why these formulas are true. There is an amusing legend in which the great mathematician Gauss, at age 10, was required by his teacher to sum up the numbers from 1 to 100. The teacher was astonished when he was able to do this in a fraction of a minute by grouping the numbers in 50 pairs that each added to 101. This easily generalizes to an easily understandable and intuitively clear proof of the identity for the sum of the first n integers
On the other hand, the sum of cubes identity can be derived by adding up the number of rectangles on a square chessboard in two different ways.
The sum of squares identity
is a different matter. There are only two kinds of proofs for this identity in calculus textbooks, one using induction and the other involving a telescoping sum of cubes. I couldn't find a single textbook that gave a proof different from these two. Neither of these gives much of a feel for why the formula looks like it does. So I set out on a quest to look for a more satisfactory explanation. I was surprised to find a wide range of proofs of this formula--at least a dozen--but I had to search for a long time before encountering any demonstration that made the identity truly obvious. I ultimately did find two such explanations. Stay tuned for the later addition of descriptions for these and other methods of proving the sum of squares formula.
Meanwhile, here is the PowerPoint presentation from one of my talks, "Intuitively clearer proofs of the sum of squares formula", on this topic.
Below is an abstract for the talk.
All modern calculus textbooks include formulas for the sum of the first n integers and the sums of their squares and cubes. These formulas are indispensable in evaluating integrals as limits of Riemann sums. The sum of integers formula can be explained in such a way that it is immediately obvious that the formula is correct. Why is the sum of squares formula valid? Textbooks that prove it do so using induction or an argument involving a telescoping sum with cubic terms, neither of which make very clear why the formula has the form that it does. In this talk we explore various alternative derivations of the sum of squares formula. While none of these seem to be quite as illuminating as the proof of the sum of integers formula, they offer an improvement over the standard arguments.
Several quite illuminating proofs were later found and included in this talk.
Later I will include more information and links on this page.