Question

If $S=i+2i^2+3i^3+...$ up to 200 term, then S equals

• A. $200i$
• B. $100(1+i)$
• C. $100(1-i)$
• D. $200(1-i)$

Answer 1

The correct option is C $100(1-i)$

$S=i+2i^2+3i^3+...200i^{200}$
$S\left(i\right)=i^2+2i^3+...\left(199\right)i^{200}+200i^{201}$
$S(1-i)=i+i^2+i^3+...i^{200}-200i^{201}$
$S(1-i)=\frac{i(1-i^{200})}{1-i}-200i^{201}$
$S(1-i)=-200\left(i\right)$
$S=\frac{-200i}{1-i}$
$S=\frac{-200\left(i\right)(i+1)}{2}$
$=-100i(i+1)$
$=100(1-i)$