The following question is present in the IMO shortlist 2013:
"Let n be a positive integer and let $a_1, . . . , a_{n-1}$ be arbitrary real numbers. Define the sequences $$u_0, . . . , u_n$$ and $$v_0, . . . , v_n$$ inductively by $$u_0 = u_1 = v_0 = v_1 = 1$$, and $$u_{k+1}= u_k + a_ku_{k-1}, v_{k+1}= v_k + a_{n-k}v_{k-1}$$ for $k= 1, . . . , n-1$. Prove that $$u_n = v_n$$
Among several solutions to this questions, there seems to exist a solution making use of continued fractions. The official Shortlist document here https://www.imo-official.org/problems/IMO2013SL.pdf mentions it in A1 comment 2.
However I am unable to provide a solution using continued fractions. Please help!
Also, I would like to know some good references to learn continued fractions