Define $a_1 = 1$ and for all natural $n$'s, $a_{n+1} = 1 + \dfrac{1}{a_n}$.
Prove that for every natural $n$, $$a_n = \dfrac{F_{n+1}}{F_n}.$$
I'm not sure if this is an induction problem or not, but could someone help me understand what is going on?