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# If $\alpha$ is $n^{th}$ root of unity,the value of $1+2\alpha+3\alpha^2+...$ upto n terms is equal to

$\begin{array}{1 1}(A)\;-\large\frac{2n}{(1-\alpha)^2}\\(B)\;-\large\frac{n}{(1-\alpha)}\\(C)\;-\large\frac{2n}{(1-\alpha)}\\(D)\;-\large\frac{n}{(1-\alpha)^2}\end{array}$

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

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