# 1992 AHSME Problems/Problem 15

## Problem

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 $z_{111}$? $\text{(A) } 1\quad \text{(B) } \sqrt{2}\quad \text{(C) } \sqrt{3}\quad \text{(D) } \sqrt{110}\quad \text{(E) } \sqrt{2^{55}}$

## Solution $\fbox{B}$ 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 $\sqrt{(-1)^2+1^2} = \sqrt{2}$.

## See also

 1992 AHSME (Problems • Answer Key • Resources) Preceded byProblem 14 Followed byProblem 16 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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