Limit of ratios of consecutive Fibonacci terms

Let $F_n$ denote the Fibonacci numbers. The limit of $F_{n+1}/F_n$ as $n\to\infty$ can be written as $\dfrac{a+\sqrt b}{c}$ with $a,b,c$ pairwise coprime positive integers. What is $abc$?

10. Divide the recurrence by the middle term: the ratio $r_n=F_{n+1}/F_n$ obeys $r_n=1+1/r_{n-1}$, so its limit $L$ solves $L^2=L+1$. The positive root is the golden ratio $\tfrac{1+\sqrt5}{2}$, so $abc=1\cdot5\cdot2=10$.