# Find the value of $\sum_{n=1}^{200}i^n$

$\begin{array}{1 1}(A) \;0\\(B)\;1\\(C)\;i\\(D)\;200\end{array}$

Toolbox:
• $i^4=i^8=i^{12}=.........i^{4n}=1$
• $1+i+i^2+i^3=0$
$\sum_{n=1}^{200}i^n=i+i^2+i^3+.............i^{200}$
$=(i+i^2+i^3+i^4)+i^4(i+i^2+i^3+i^4)+i^8(i+i^2+i^3+i^4)+......$
$=(i+i^2+i^3+1)+1(i+i^2+i^3+1)+...........$
=0
edited Jul 28, 2014