Question

Decide whether the starting value z0=cis3π/17 will eventually result in cyclic behavior and explain your reasoning. If so, describe the n-cycle.

Answer 1

these are the results i have obtained, i can see that it cycles but i dont know how to describe it, the cycle never goes back to z0

Answer 2

\[\large {z_0^{18}=cis 18(\frac{3\pi}{17})=cis\frac{54\pi}{17}=cis(3\pi+\frac{3\pi}{17})\\\qquad =cis3\pi\cdot cis\frac{3\pi}{17}=-cis\frac{3\pi}{17}\\z_0^{17}\cdot z_0^{18}=cis3\pi\cdot (-cis\frac{3\pi}{17})=cis\frac{3\pi}{17}}\]

Answer 3

also, if it has a cyclic behavior, then \[\large z_0^{n+1}=z_0\]
or equivalently, \[\large z_0^{n}=1\]

solving the equation above, \[\large {z_0^n=1\\cis n(\frac{3\pi}{17})=1\Rightarrow \frac{3n\pi}{17}=2k\pi}\]

the least positive value of \(n\) that will make \[\large \frac{3n}{17}\] even is \[n=34\]