IMO Shortlist 1984 problem 6
Dodao/la:
arhiva2. travnja 2012. Let
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
be a positive integer. The sequence
![\{f_n\}](/media/m/b/3/2/b32f7aaac7fb991c47e212c9c1ba3f12.png)
is defined as follows:
![f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).](/media/m/1/4/f/14fdd14265658c0f6c6b04f6dc5cf2a9.png)
Show that for each
![k \in \mathbb N](/media/m/4/4/1/4414841501431d4eaa4e6cf3b7c2d9c9.png)
there exists
![r \in \mathbb N](/media/m/4/c/3/4c3f03292c5f48331c08795415ca39cd.png)
such that
%V0
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
$$f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).$$
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$
Izvor: Međunarodna matematička olimpijada, shortlist 1984