Question

The sum of n terms of $\mathrm{1.2.3}+\mathrm{2.3.4}+\mathrm{3.4.5}+...$ is

Answer 1

$1\cdot 2\cdot 3+2\cdot 3\cdot 4+3\cdot 4\cdot 5+\dots$

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

$=(n^2+n)(n+2)$

${=n}^3+{2n}^2+n^2+2n$

${=n}^3+{3n}^2+2n$

$S_n={\sum }_{n=1}^{\infty }{T}_n$

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

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

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

$S_n=\frac{n(n+1)}{4}[n^2+n+4n+2+4]$

$S_n=\frac{n(n+1)}{4}[n^2+5n+6]$