Question

lf $\alpha$ be the $n^{th}$ root of unity then the sum of the series $1+2\alpha +3{\alpha }^2+\dots .+n{\alpha }^{n-1}$ equals?

• A. $\frac{-n}{1-\alpha }$
• B. $\frac{-n}{(1-\alpha {)}^2}$
• C. $\frac{n}{(1-\alpha )}$
• D. $\frac{n}{(1-\alpha {)}^2}$

Answer 1

The correct option is A $\frac{-n}{1-\alpha }$

$s=1+2\alpha +3{\alpha }^2--n{\alpha }^{n-1}$
$s\alpha =\alpha +2{\alpha }^2+3{\alpha }^3--n\alpha So,s-s\alpha =1+\alpha +{\alpha }^2--{\alpha }^{n-1}-n{\alpha }^n$
(Sum of roots of unity)
$\Rightarrow s(1-\alpha )=-n{\alpha }^n$
$\Rightarrow s=\frac{-n{\alpha }^n}{1-\alpha }=\frac{-n}{1-\alpha }$
as ${\alpha }^n=1$
$\alpha$ being of unity root