Question

If $\alpha$ be the nth root of unity then the value of $1+2\alpha +3{\alpha }^2+....$ to $n$ term's equals

• A. $\frac{n}{1-\alpha }$
• B. $\frac{n+1}{\alpha -1}$
• C. $\frac{n}{1+\alpha }$
• D. $\frac{-n}{1-\alpha }$

Answer 1

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

Let $S=1+2\alpha +3{\alpha }^2+...+n{\alpha }^{n-1}....\left(i\right)$
$\therefore \alpha S=\alpha +2{\alpha }^2+....(n-1){\alpha }^{n-1}+n{\alpha }^n...\left(ii\right)$
subtracting (ii) from (i) we get
$S(1-\alpha )=\frac{1-{\alpha }^n}{1-\alpha }-n{\alpha }^n$
$S(1-\alpha )=-n{\alpha }^n$
$S=\frac{-n}{1-\alpha }(\sqrt[n]{n1}=\alpha \therefore {\alpha }^n=1)$