Question 79
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of integers and $b_{0}, b_{1}, b_{2}, \ldots$ be a sequence of positive integers such that $a_{0}=0$, $a_{1}=1$, and
$$
a_{n+1}=\left\{\begin{array}{ll}
a_{n} b_{n}+a_{n-1}, & \text{ if } b_{n-1}=1 \\
a_{n} b_{n}-a_{n-1}, & \text{ if } b_{n-1} \gt 1
\end{array} \quad \text{ for } n=1,2, \ldots\right.
$$
Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.
$$
a_{n+1}=\left\{\begin{array}{ll}
a_{n} b_{n}+a_{n-1}, & \text{ if } b_{n-1}=1 \\
a_{n} b_{n}-a_{n-1}, & \text{ if } b_{n-1} \gt 1
\end{array} \quad \text{ for } n=1,2, \ldots\right.
$$
Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.