Standard Induction Hint

Standard Induction Hint (This is the same hint for all induction problems):

Let $S$ be the set of natural numbers for which the statement $1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}$ is true. We want to show that $S=\mathbb{N}$.

1. First show that the statement is true if you let $n=1$. (Show $1\in S$.)
2. Then assume that the statement is true if you let $n=k$ for some $k\in \mathbb{N}$.
3. Use this assumption to then prove that the statement is true when you let $n=k+1$. (This shows if $k\in S$, then $k+1\in S$.)

You can then apply the principle of mathematical induction to claim $S=\mathbb{N}$.