Determine the value of $\sum_{k=0}^n i^k$ this is the problem:

Determine the value of the sum $\sum_{k=0}^n i^k$ for $n \in \Bbb N$.

I started to sum term by term but don't get its value 

$S = i^0 + i^1 + i^2 + ... + i^{n-1} + i^n$
$Si = i^1 + i^2 + i^3 + ... + i^n + i^{n+1}$
$Si - S = i^{n+1} - i^0$
$S(i-1) = i^{n+1} - i^0$
$S = \dfrac{i^{n+1} - 1}{i-1}$
