Question 162
Let $f(x) = c_m x^m + c_{m-1} x^{m-1} + \cdots + c_1 x + c_0$, where each $c_i$ is a nonzero integer. Define a sequence $\{a_n\}$ by $a_1 = 0$ and $a_{n+1} = f(a_n)$ for all positive integers $n$.
a. Let $i$ and $j$ be positive integers with $i \lt j$. Show that $a_{j+1} - a_j$ is a multiple of $a_{i+1} - a_i$.
b. Show that $a_{2008} \neq 0$.
a. Let $i$ and $j$ be positive integers with $i \lt j$. Show that $a_{j+1} - a_j$ is a multiple of $a_{i+1} - a_i$.
b. Show that $a_{2008} \neq 0$.