Limit of Fibonacci terms two steps apart

With $F_n$ the Fibonacci numbers, the limit of $F_{n+2}/F_n$ as $n\to\infty$ can be written as $\dfrac{a+\sqrt b}{c}$ with $a,b,c$ positive integers that are pairwise coprime. What is the product $abc$?

30. Since $F_{n+2}=F_{n+1}+F_n$, the ratio equals $F_{n+1}/F_n+1\to\varphi+1=\varphi^2=\tfrac{3+\sqrt5}{2}$, so $abc=3\cdot5\cdot2=30$.