2013 IMO Shortlist A1

Let $n$ be a positive integer and let $a_{1}, \ldots, a_{n-1}$ be arbitrary real numbers. Define the sequences $u_{0}, \ldots, u_{n}$ and $v_{0}, \ldots, v_{n}$ inductively by $u_{0}=u_{1}=v_{0}=v_{1}=1$, and $$u_{k+1}=u_{k}+a_{k} u_{k-1}, \quad v_{k+1}=v_{k}+a_{n-k} v_{k-1} \quad \text { for } k=1, \ldots, n-1 .$$ Prove that $u_{n}=v_{n}$. (France)

令$n$ 为正整数，并令$a_{1}, \ldots, a_{n-1}$ 为任意实数。通过 $u_{0}=u_{1}=v_{0}=v_{1}=1$ 和 $$ u_{k+1}=u_{k}+a_{k} u_{k-1}, \quad 归纳定义序列 $u_{0}, \ldots, u_{n}$ 和 $v_{0}, \ldots, v_{n}$ v_{k+1}=v_{k}+a_{n-k} v_{k-1} \quad \text { for } k=1, \ldots, n-1 。 $$证明$u_{n}=v_{n}$。 (法国)