Is there a system of equations model to solve sum of squared integers?

Question

I have made a wooden block model (5x5,4x4...1x1) to model geometrically "sum of squared integers" for my students to examine. aside from using plug-n-play with the standard formula given from our textbooks $\sum_{i=1}^n i^2 = \frac{1}{6}n(n+1)(2n+1)$, is there a "system of equations" that we could use to also model/solve sum of squared integers graphically?

Answer 1

If you just want the leading term of the sum and you assume your solution is a sum of weighted powers of $n$ (which is indeed the case), then you can note that as $n \to \infty$, the volume of the pyramid given by stacking a $1 \times 1$ square of unit cubes on top of a $2 \times 2$ square of unit cubes etc. until you reach a base of $n \times n$ square of unit cubes approaches the volume of a standard square pyramid with height $n$ and base side length $n$, which has volume $n^3/3$. So your sum is $n^3/3$ plus lower order terms.