Find a formula for $0 · 1 · 2 + 1 · 2 · 3 + 2 · 3 · 4 + \dots +n(n + 1)(n + 2)$, for $n \in \mathbb N$ $$\sum\limits_{i=1}^n i(i + 1)(i + 2)$$
$$\sum\limits_{i=1}^n i^3 + 3i^2 + 2i$$
$$\sum\limits_{i=1}^n i^3 + 3\sum\limits_{i=1}^ni^2 + 2\sum\limits_{i=1}^ni$$
$$= (\frac14)n^4 + (\frac12)n^3 + (\frac14)n^2 + n^3 + 3(\frac{n^2}2) + (\frac{n}2) + n^2 + n$$
$$ =n^4 + 3n^3 + 8n^2 + 3n$$
Why does what I did above not work?

It is very much easier to note that $$n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2)=4n(n+1)(n+2)$$ from which you get a telescoping series.
This is related to the binomial identity $\binom nr+\binom n{r+1}=\binom {n+1}{r+1}$ with $r=3$, and the pattern clearly generalises.
