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.