Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer and let
![a_1, \ldots, a_{n-1}](/media/m/d/3/d/d3dddd1a0720ebf80a8d28d80ec8181c.png)
be arbitrary real numbers. Define the sequences
![u_0, \ldots, u_n](/media/m/1/0/a/10a068541ec6936fb4c5c692d9f64b99.png)
and
![v_0, \ldots, v_n](/media/m/8/9/b/89bb37acf901adbf97e398de1875c181.png)
inductively by
![u_0 = u_1 = v_0 = v_1 = 1](/media/m/4/d/7/4d79d078d878b3a6fa608b6f583f875b.png)
, and
![u_{k+1} = u_k + a_k u_{k-1}, \quad v_{k+1} = v_k + a_{n-k} v_{k-1}](/media/m/8/3/1/831cd7f6afeb494a44a2142865cb3643.png)
for
![k = 1, \ldots, n - 1](/media/m/b/e/d/bed1390f8bd6604fc88ca16e9573aac6.png)
.
Prove that
![u_n = v_n](/media/m/b/2/0/b20ef07c6e6c8409e81fe393a2a32634.png)
.
%V0
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 $$
u_{k+1} = u_k + a_k u_{k-1}, \quad v_{k+1} = v_k + a_{n-k} v_{k-1}
$$ for $k = 1, \ldots, n - 1$.
Prove that $u_n = v_n$.