Question 30
Let $a_0, a_1, a_2, \dots$ be a sequence of integers and $b_0, b_1, b_2, \dots$ a sequence of positive integers such that $a_0 = 0, a_1 = 1$, and for all $n = 1, 2, \dots$,
$$
a_{n+1} = \begin{cases} a_n b_n + a_{n-1} & \text{if } b_{n-1} = 1; \\ a_n b_n - a_{n-1} & \text{if } b_{n-1} \gt 1. \end{cases}
$$
Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ is greater than or equal to 2017.
$$
a_{n+1} = \begin{cases} a_n b_n + a_{n-1} & \text{if } b_{n-1} = 1; \\ a_n b_n - a_{n-1} & \text{if } b_{n-1} \gt 1. \end{cases}
$$
Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ is greater than or equal to 2017.