benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / The 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 i…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

The sequence (zn)\left(z_{n}\right) of complex numbers satisfies the following properties: z1z_{1} and z2z_{2} are not real. zn+2=zn+12znz_{n+2}=z_{n+1}^{2} z_{n} for all integers n≥1n \geq 1. zn+3zn2\frac{z_{n+3}}{z_{n}^{2}} is real for all integers n≥1n \geq 1. |z3z4|=|z4z5|=2\left|\frac{z_{3}}{z_{4}}\right|=\left|\frac{z_{4}}{z_{5}}\right|=2 Find the product of all possible values of z1z_{1}.
Plain-text mathematical notation (without MathML)
The 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
The sequence $\left(z_{n}\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \geq 1$. $\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \geq 1$. $\left|\frac{z_{3}}{z_{4}}\right|=\left|\frac{z_{4}}{z_{5}}\right|=2$ Find the product of all possible values of $z_{1}$.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
8ed39eb6-d242-529c-a8f3-d694ad76fac9
task_key
test--8ed39eb6-d242-529c-a8f3-d694ad76fac9
task_revision_id
1
config / split
/ test

Official source

initial import