Question 14Let a sequence be defined as follows: $a_1 = 3,$ $a_2 = 3,$ and for $n \ge 2,$ $a_{n+1}a_{n-1} = a_n^2 + 2007.$ Find the largest integer less than or equal to $\frac{a_{2007}^2 + a_{2006}^2}{a_{2007}a_{2006}}.$