Question

The sum of the series $i+2i^2+3i^3+...$ up to $200$ terms equals

• A. $100(i-1)$
• B. $100(1-i)$
• C. $200i$
• D. None of these

Answer 1

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

$S=i+{2i}^2+{3i}^3+......+200i^{200}$ ...(1)
$iS=i^2+{2i}^3+......+199i^{200}+200i^{201}$
Subtracting (1) & (2), we get
$S(1-i)=i+i^2+i^3+......+i^{200}-200i^{201}\Rightarrow S(1-i)=\frac{i(1-i^{200})}{1-i}-200i\Rightarrow S(1-i)=-200i\Rightarrow S=\frac{-200i}{1-i}=100(1-i)$
Ans: B