Question

If $\alpha$ is one of the $n^{\text{th}}$ complex root of unity other than unity, then the sum of $n$ terms of the series $1+2\alpha +3{\alpha }^2+.....$ is given by $\frac{pn}{1+q\alpha }$, then the value of $p^2+q^2$ is

Answer 1

$S=1+2\alpha +3{\alpha }^2+......+n{\alpha }^{n-1}\alpha S=0+\alpha +2{\alpha }^2+3{\alpha }^3+...+(n-1){\alpha }^{n-1}+n{\alpha }^n$

Subtracting above equations
$S(1-\alpha )=1+\alpha +{\alpha }^2+....+{\alpha }^{n-1}-n{\alpha }^n\Rightarrow S(1-\alpha )=0-n\cdot 1\Rightarrow S=\frac{-n}{1-\alpha }\Rightarrow p=-1,q=-1$