Let $i=\sqrt{-1}$. Define a sequence of complex numbers by
\[z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.\]
In the complex plane, how far from the origin is $z_{111}$?
$\text{(A) } 1\quad \text{(B) } \sqrt{2}\quad \text{(C) } \sqrt{3}\quad \text{(D) } \sqrt{110}\quad \text{(E) } \sqrt{2^{55}}$
$\fbox{B}$ Write out some terms: $0, i, -1+i, -i, -1+i, -i, -1+i, -i$, etc., and it keeps alternating between $-1+i$ and $-i$, so as $111$ is odd, $z_{111}$ is $-1+i$. Thus its distance from the origin is $\sqrt{(-1)^2+1^2} = \sqrt{2}$.