How would one prove that
$$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$
where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?
Limits – How to Prove $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
fibonacci-numbersgolden ratiolimits
Best Answer
$$F_{n+1}=F_n+F_{n-1}$$
$$F_{n+1}/F_n=1+F_{n-1}/F_n=1+1/(F_n/F_{n-1})$$
Call the limit $x$; then $$x=1+1/x$$
Take it from there.