If  $\alpha$ is $n^{th}$  root of unity, then $1+2\alpha +3{\alpha }^2+\mathrm{.......}+n{\alpha }^{n-1}=$

$\begin{array}{l}{\alpha }^n=1givenlets=1+2\alpha +3{\alpha }^2+--+n{\alpha }^{n-1}\\ \frac{\alpha s=\_\alpha +\_2{\alpha }^2+\_3{\alpha }^3+\_n{\alpha }^n}{\Rightarrow \left(1-\alpha \right)s=1+\alpha +{\alpha }^2+{\alpha }^3----{\alpha }^{n-1}-n{\alpha }^n}\\ \Rightarrow \left(1-\alpha \right)s=\left(\frac{1-{\alpha }^n}{1-\alpha }\right)-n\left(1\right)\\ \Rightarrow \left(1-\alpha \right)s=\frac{1-1}{1-\alpha }-n\\ \Rightarrow \left(1-\alpha \right)s=-n\\ \Rightarrow s=\frac{-n}{1-\alpha }\end{array}$