SMO 2009 Open Round 1 Question 16

Let $(a_n)$ be positive integers with $a_1=1$, $a_2=2009$, and $a_{n+2}a_n-a_{n+1}^2-a_{n+1}a_n=0$ for $n\ge1$. Determine $\dfrac{a_{993}}{100a_{991}}$.