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MATH / A sequence (z_(n)) of complex numbers satisfies the following properties:

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problem

A sequence (zn)(z_n) of complex numbers satisfies the following properties: z1z_1 and z2z_2 are not real. zn+2=zn+12znz_{n+2}=z_{n+1}^2z_n for all integers n≥1n\geq 1. zn+3zn2\dfrac{z_{n+3}}{z_n^2} is real for all integers n≥1n\geq 1. |z3z4|=|z4z5|=2\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2. Find the product of all possible values of z1z_1.
Plain-text mathematical notation (without MathML)
A sequence (z_(n)) of complex numbers satisfies the following properties:

z₁ and z₂ are not real.
z_(n+2)=z_(n+1)²z_(n) for all integers n≥1.
(z_(n+3))/(z_(n)²) is real for all integers n≥1.
|(z₃)/(z₄)|=|(z₄)/(z₅)|=2.

Find the product of all possible values of z₁.
Original LaTeX notation
A 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 $z_1$.

level

Level 5

type

Precalculus

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Source and history

task_id
315c4f83-0ad5-55dd-9fc3-6c9f54b6463a
task_key
precalculus--test--315c4f83-0ad5-55dd-9fc3-6c9f54b6463a
task_revision_id
1
config / split
precalculus / test

Official source

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