Question

Find the sum of the series up to $n$ terms ;$1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+............$

Answer 1

Find the sum of series up to n term

$1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+$ _ _ _

$={\sum }_{n=1}^{n}n(n+1)(n+2)$

$={\sum }_{n=1}^n(n^2+n)(n+2)$

$={\sum }_{n=1}^n{n}^3+2n^2+n^2+2n$

$={\sum }_{n=1}^n{n}^3+3n^2+2n$

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

$=\frac{6n^2(n+1{)}^2+12n(n+1\left)\right(2n+1)+48n}{24}$

$=\frac{1}{24}\left(6n^2\right(n+1{)}^2+12n(n+1)(2n+1)+48n)$