Question 97
Consider the polynomial $f(x) = c x(x - 2)$ where $c$ is a positive real number. For any $n \in \mathbb{Z}^{+}$, the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers.
1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once.
2. Prove that $c \ge 1$.
1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once.
2. Prove that $c \ge 1$.