Question 58
Consider a complex number $z$, $z \neq 0$ and the real sequence
$$
a_n = \left| z^n + \frac{1}{z^n} \right|, \quad n \ge 1.
$$
a) Show that if $a_1 \gt 2$, then
$$
a_{n+1} \lt \frac{a_n + a_{n+2}}{2}, \text{ for all } n \in \mathbb{N}^*.
$$
b) Prove that if there exists $k \in \mathbb{N}^*$ such that $a_k \le 2$, then $a_1 \le 2$.
$$
a_n = \left| z^n + \frac{1}{z^n} \right|, \quad n \ge 1.
$$
a) Show that if $a_1 \gt 2$, then
$$
a_{n+1} \lt \frac{a_n + a_{n+2}}{2}, \text{ for all } n \in \mathbb{N}^*.
$$
b) Prove that if there exists $k \in \mathbb{N}^*$ such that $a_k \le 2$, then $a_1 \le 2$.