2018 Putnam B4

Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_n x_{n-1} - x_{n-2}$ for $n \geq 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.

给定实数 $a$，我们通过 $x_0 = 1$、$x_1 = x_2 = a$ 和 $x_{n+1} = 2x_n x_{n-1} - x_{n-2}$ 定义一个序列，其中 $n \geq 2$。证明如果对于某些 $n$，$x_n = 0$，则该序列是周期性的。