Problem:
Let i=−1​. Define a sequence of complex numbers by z1​=0,zn+1​=zn2​+i for n≥1. In the complex plane, how far from the origin is z111​?
Answer Choices:
A. 1
B. 2​
C. 3​
D. 110​
E. 255​
Solution:
We compute z1​=0,z2​=i,z3​=i−1,z4​=−i, and z5​=i−1. Since z5​=z3​, it follows that z111​=z109​=z107​=⋯=z5​=z3​=i−1, which is 2​ units from the origin.
Note. The Mandelbrot set is defined to be the set of complex numbers c for which all the terms of the sequence defined by z1​=0,zn+1​=zn2​+c for n≥1, stay close to the origin. Thus c=i is in the Mandelbrot set.