MathDB

2013 HMMT Guts #25: Sequence of Complex Numbers

Source:

March 26, 2013

HMMTcomplex numbers

Problem Statement

The sequence

(z_n)

of complex numbers satisfies the following properties:
[*]

z_1

and

z_2

are not real. [*]

z_{n+2}=z_{n+1}^2z_n

for all integers

n\geq 1

. [*]

\dfrac{z_{n+3}}{z_n^2}

is real for all integers

n\geq 1

. [*]

\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2

. Find the product of all possible values of

z_1