If  $n\ge 2$ is a positive integer, then the sum of the series ${ }^{n+1}{C}_2+2\left({ }^2{C}_2{+}^3{C}_2{+}^4{C}_2+{\mathrm{..............}}^n{C}_2\right)$ is

$\begin{array}{c}{ }^{n+1}{C}_2+2\left({ }^2{C}_2{+}^3{C}_2{+}^4{C}_2+\mathrm{............}{+}^n{C}_2\right){=}^{n+1}{C}_2+2\left({ }^{n+1}{C}_3\right)\\ {=}^{n+1}{C}_2{+}^{n+1}{C}_3{+}^{n+1}{C}_3\\ { }^{n+2}{C}_3{+}^{n+1}{C}_3=\frac{\left(n+2\right)\left(n+1\right)\left(n\right)}{6}+\frac{\left(n+1\right)·n\left(n-1\right)}{6}\\ =\frac{\left(n+1\right)n}{6}\left(2n+1\right)\end{array}$