Question

Find the sum to $n$ terms of the series:
$1\times 2+2\times 3+3\times 4+4\times 5+\cdots$

Answer 1

$S=1\times 2+2\times 3+3\times \mathrm{4.......}n(n+1)S=\sum _{r=1}^{r=n}r(r+1)S=\sum _{r=1}^{r=n}r(r+1)S=\sum _{r=}^n{r}^2+rS=\sum _{r=}^n{r}^2+\sum _{r=}^{n}rS=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}S=\frac{2n^3+3n^2+n}{6}+\frac{n^2+n}{2}$

$S=\frac{2n^3+6n^2+4n}{6}$