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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} $

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$S=1+2\alpha+3\alpha^2+.....+n\alpha ^{n-1}$
$\alpha S=\alpha+2\alpha^2+.......(n-1)\alpha^{n-1}+n\alpha ^n$
$S-\alpha S=1+\alpha+\alpha^2+.....+\alpha^{n-1}-n\alpha^n$
$\Rightarrow (1-\alpha)S=\large\frac{1-\alpha^n}{1-\alpha}$$-n\alpha^n$
$\Rightarrow -n$($\alpha^n=1)$
$\Rightarrow S=-\large\frac{n}{1-\alpha}$
Hence (B) is the correct answer.
answered Apr 10, 2014 by sreemathi.v
 

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