let :${\displaystyle F_{1}=F_{2}=1,\forall n>2:F_{n}=F_{n-1}+F_{n-2}}$ then :

Question

let :${\displaystyle F_{1}=F_{2}=1,\forall n>2:F_{n}=F_{n-1}+F_{n-2}}$

then :

$$\frac{F_{100}}{F_{99}}+\frac{F_{99}}{F_{100}}=?$$

my try :

$1=\frac{F_{n-1}}{F_{n}}+\frac{F_{n-2}}{F_{n}}!!!!$

Continuation!!!

Answer 1

We have $$ \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi = \frac{1 + \sqrt{5}}{2} $$ Therefore, $$ \frac{F_{n+1}}{F_n} + \frac{F_n}{F_{n+1}} \approx \varphi + \frac{1}{\varphi} = \sqrt{5} $$ Indeed, $$ \frac{F_{100}}{F_{99}}+\frac{F_{99}}{F_{100}} - \sqrt{5} \approx -5.8 \times 10^{-42} $$