Let i=√-1. For n≥1, define a sequence of complex numbers by z_1&=0, z_n+1&=z_n^2+i. In the complex plane, how far from the origin is z_111?

1992 AMC 12 Problem 15

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15.

Let

i=\sqrt{-1}.

For

n\ge1,

define a sequence of complex numbers by

\begin{aligned} z_1&=0,\\ z_{n+1}&=z_n^2+i. \end{aligned}

In the complex plane, how far from the origin is

z_{111}?

1

\sqrt2

\sqrt3

\sqrt{110}

\sqrt{2^{55}}

Compute the first few terms rather than trying to expand

z_{111}

Look for a cycle beginning with

z_3

The first terms are

\begin{aligned} z_1&=0,\quad z_2=i,\quad z_3=-1+i,\\ z_4&=-i,\quad z_5=-1+i. \end{aligned}

Hence the sequence alternates between

-1+i

at odd indices and

-i

at even indices from

z_3

onward. Therefore

z_{111}=-1+i,

whose distance from the origin is

\sqrt{(-1)^2+1^2}=\sqrt2.

Thus the correct answer is B.