The sum of the first n squares

A while ago, I spoke to one of my Year 7 groups, and showed them that there was a simple formula for the sum of the first n integers.

I also pointed out that there was a formula for the sum of the first n squares.

I gave a clue. A short while later, Sofia had found the formula. Here is a link to her work.

The sum of the first n integers, and the first n squares.

As an addition to Sofia's splendid work (unprecedented for a Year 7, in my experience), here is an inductive proof of her formula.

An Inductive Proof