1992 AHSME Problem 15

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}}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.