2017 IMO Shortlist A7

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 $$a_{n+1}=\left\{\begin{array}{ll} a_{n} b_{n}+a_{n-1}, & \text { if } b_{n-1}=1 \\ a_{n} b_{n}-a_{n-1}, & \text { if } b_{n-1}>1 \end{array} \quad \text { for } n=1,2, \ldots\right.$$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to 2017 . (Australia)

设 $a_{0}, a_{1}, a_{2}, \ldots$ 为整数序列，$b_{0}, b_{1}, b_{2}, \ldots$ 为正整数序列，使得 $a_{0}=0, a_{1}=1$ 和 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}>1 \end{array} \quad \text { for } n=1,2, \ldots\right。 证明 $a_{2017}$ 和 $a_{2018}$ 两个数字中至少有一个必须大于或等于 2017 。 (澳大利亚)