ILL 1984 Problem 14

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: $$f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 (n \geq 2).$$

Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$