<s>[INST] $n^2 + (n-1)^2 + \dots +2^2 + 1^2$ equals what? $$n^2 + (n-1)^2 + \dots +2^2 + 1^2 = \text{??}$$
I am reading that this equals 
$$\frac{1}{3} n \left( n + \frac{1}{2} \right) (n+1) $$
But have no clue how.. The thing that strikes me most is the fact that the latter has a cubed factor when expanded (i.e. in big O notation it's O$(n^3)$). 
 [/INST] Hint: Use induction on $n$
$$n=1:\quad1^2=\frac{1}{3}(1)\left(\frac{3}{2}\right)(2)=1$$
Assume statement is true for $n$, then
\begin{align*}
(n+1)^2+n^2+\dots+2^2+1^2&=n^2+2n+1+\frac{1}{3}(n)\left(n+\frac{1}{2}\right)(n+1)\\
\end{align*}
Then expand and factorise and you'll get your answer.
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