Looking at sums of integers and squares via geometry and algebra

Earlier this week I carelessly asked my younger son to read a section in a statistics book that relied on knowledge of calculus:

My son had some interesting questions about ideas in calculus, so instead of statistics we spent a few days this week talking about calculus (mainly finding the area under a curve).

For our project today I wanted to revisit one of the ideas in arithmetic that we’d relied on in the calculus discussion – sums of integers and sums of squares.

We started with sums of integers and my son gave a geometric proof of the rule for sums of consecutive integers:

Now we moved on to sums of squares – here we talked about the sum (and some of the number theory hiding in the sum formula), but didn’t yet try to prove the result:

Now we looked at a geometric way to understand the sum of squares formula:

Finally, we did a lightning fast review of mathematical induction and showed how we could proved that the sum of squares formula was true in general:

So, a fun week and a bunch of great discussions . . . even if it started with me being pretty careless!

One thought on “Looking at sums of integers and squares via geometry and algebra

  1. It just occurred to me how to prove int x^2 = 1/3 x^3 for somebody pre-calculus. Decide that x^2 = area of a square, and that the question is about volume of the polyhedron, a cone on a square. Then figure out that three of those fit into a cube.

    For the sums of squares thing: I think the coolest approach to such problems is to decide that monomials x^k are just bad when thinking about such functions, and (x choose k) is the way to go. Then when you have any sequence, you can look at successive differences, and figure out the coefficients when expanding into (x choose k)s. What’s fun is that the resulting Taylor’s-theorem-like formula works for any function NN -> RR at all! (The infinite sum is finite when evaluated at any natural.) In cases like the sum-of-squares, though, only finitely many coefficients are nonzero.
    Sequence: sums of squares 0,1,5,…
    First difference: 1,4,9,16,25,36,…
    Second difference: 3,5,7,9,…
    Third difference: 2,2,2,2,…
    Fourth and on: 0,0,0,0
    Hence the sum of the first x squares is 0(x choose 0) + 1(x choose 1) + 3(x choose 2) + 2(x choose 3), or x + 3x(x-1)/2 + 2x(x-1)(x-2)/6 if you insist on working with monomials and their ugly rational coefficients.

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