<s>[INST] All sequences constructed by using the denominator as nominator and the sum of denominator and nominator as denominator converges to $\phi-1$ Assume we are given any number a. Write it in the form $a = \frac{b}{c}$ (if rational, in the usual way, if irrational, use forms like $\frac{a}{1}$).
Construct a sequence $\frac{b}{c},\frac{c}{b+c},\frac{b+c}{b+2 c},\frac{b+2c}{2b+3c}...$
The nth term is $\frac{F_{n-1}*b+F_n*c}{F_n*b+F_{n+1}*c}$ where $F_n$ is the nth Fibonacci number.
I have noticed that no matter what the starting point $a$ is, the sequence always converges to $\phi-1$ , ie. $\frac{(\sqrt{5}+1)}{2}-1$ or $\frac{(\sqrt{5}-1)}{2}$.
I would like to prove this. If $a=1$, then the nth term is $F_{n+1}/F_{n+2}$ and the limit is $\phi-1$. But If $a \not= 1$ this seems harder to show. Especially if $a < 0$.
 [/INST] Given $\displaystyle\frac{F_{n+1}}{F_n}\to\varphi$ and $\displaystyle\frac{F_{n-1}}{F_n}\to\varphi^{-1}$, your sequence converges as
$$\frac{1/F_n}{1/F_n}\cdot\frac{F_{n-1}b+F_nc}{F_nb+F_{n+1}c}=\frac{b\displaystyle\frac{F_{n-1}}{F_n}+c}{\displaystyle b+c\frac{F_{n+1}}{F_n}}\longrightarrow \frac{b\varphi^{-1}+c}{b+c\varphi}=\varphi^{-1}\frac{b+c\varphi}{b+c\varphi}=\varphi^{-1}.$$
Note that $\varphi^{-1}=\varphi-1$.
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