Question 6
Let $n$ be a positive integer and $a_1,\ldots,a_{n-1}$ arbitrary real numbers. Define $u_0,\ldots,u_n$ and $v_0,\ldots,v_n$ by $u_0=u_1=v_0=v_1=1$ and, for $k=1,\ldots,n-1$,
$$u_{k+1}=u_k+a_ku_{k-1},\qquad v_{k+1}=v_k+a_{n-k}v_{k-1}.$$
Prove that $u_n=v_n$.
$$u_{k+1}=u_k+a_ku_{k-1},\qquad v_{k+1}=v_k+a_{n-k}v_{k-1}.$$
Prove that $u_n=v_n$.