Question

Find the sum of the series upto $n$ terms $\mathrm{1.2.3}+\mathrm{2.3.4}+\mathrm{3.4.5}+...$ to $n$ terms.

Answer 1

General term$=n(n+1)(n+2)$

Sum$=\sum n(n+1)(n+2)$

$=\sum n^3+3n^2+2n$

$=\sum n^3+3\sum n^2+2\sum n$

$=\frac{n^2{(n+1)}^2}{4}+3\frac{\left[n(n+1)(2n+1)\right]}{6}+\frac{2\left[n(n+1)\right]}{2}$

$=n(n+1)\left[\frac{(n+2)(n+3)}{4}\right]$

$=\frac{n(n+1)(n+2)(n+3)}{4}$