Limit of the ratio of consecutive Fibonacci numbers as n approaches negative infinity?

Question

It is known that the lim n→∞ (+1/) is the Golden Ratio, however I'm curious to know what this limit would be if n approached negative infinity?

Answer 1

If you define $F_n$ for negative integers $n$ in such a way that $F_n=F_{n-1}+F_{n-2}$ for all integers $n$, then it’s not hard to prove by induction that $F_{-n}=(-1)^{n+1}F_n$ for all $n\in\Bbb Z^+$. It follows that

$$\lim_{n\to-\infty}\frac{F_{n+1}}{F_n}=-\lim_{n\to\infty}\frac{F_n}{F_{n+1}}=-\frac1{\varphi}=1-\varphi\;.$$