Question

Sum the following series to n terms $\mathrm{1.2.3.4}+\mathrm{2.3.4.5}+\mathrm{3.4.5.6}+....$.

Answer 1

General term $=n(n+1)(n+2)(n+3)$

$=n^4+6n^3+11n^2+6n$

Sum to $n$ terms $=\sum n^4+6\sum n^3+11\sum n^2+6\sum n$

$=[\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{3}]+6\left[\frac{n^2{(n+1)}^2}{4}\right]+11\frac{\left[n(n+1)(2n+1)\right]}{6}+\frac{6\left[n(n+1)\right]}{2}$

$=\frac{n^5}{5}+2n^4+7n^3+10n^2+\frac{9n}{2}$