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

$$\begin{gathered} S=1+2\alpha +3{\alpha }^2+\mathrm{....}+n{\alpha }^{n-1} \\ \frac{\alpha s=\alpha +2{\alpha }^2+\mathrm{....}+(n-1){\alpha }^{n-1}+n{\alpha }^n}{(-)s(1-\alpha )=1+\alpha +{\alpha }^2+\mathrm{....}+{\alpha }^{n-1}-n{\alpha }^n} \\ \sin ce {\alpha }^n=1 \Rightarrow S(1-\alpha )=0-n\left(1\right)\Rightarrow s=\frac{-n}{1-\alpha } \end{gathered}$$