Question 35
Given a natural number $n\gt 0$ and a complex number $z = x + i y$ of unit module, $x^2 + y^2 = 1$, equality may or may not be satisfied
$$
\left(z + \frac{1}{z}\right)^n = 2^{n-1}\left(z^n + \frac{1}{z^n}\right)
$$
Once $n$ is set, we will designate by $S(n)$ the subset of unit module complexes for which the given equality is satisfied. It is requested
a) Reasonably calculate $S(n)$, for $n=2,3,4,5$.
b) Upper limit the number of elements of $S(n)$ based on $n$, for $n\gt 5$.
$$
\left(z + \frac{1}{z}\right)^n = 2^{n-1}\left(z^n + \frac{1}{z^n}\right)
$$
Once $n$ is set, we will designate by $S(n)$ the subset of unit module complexes for which the given equality is satisfied. It is requested
a) Reasonably calculate $S(n)$, for $n=2,3,4,5$.
b) Upper limit the number of elements of $S(n)$ based on $n$, for $n\gt 5$.