RPS #011 - Celebrating The Best 5 National Competition

Let the limit, if it exists, be $\displaystyle \lambda = \lim_{n \to \infty} \frac {F_n}{F_{n-1}}$ . As $F_n = F_{n-1} + F_{n-2}$ , since $F_{k+1} > F_k$ for $k \ge 2$ , we have $F_{n-1} < F_n < 2F_{n-1}$ $\implies 1 < \lambda < 2$ . Since $\lambda$ is bounded, by Bolzano-Weierstrass theorem, the limit exists.

$\begin{aligned} \lambda & = \lim_{n \to \infty} \frac {F_n}{F_{n-1}} \\ & = \lim_{n \to \infty} \frac {F_{n+1}}{F_n} \\ & = \lim_{n \to \infty} \frac {F_n + F_{n-1}}{F_n} \\ & = 1 + \frac 1\lambda \end{aligned}$

$\begin{aligned} \implies \lambda^2 - \lambda - 1& = 0 \end{aligned}$

$\begin{aligned} \implies \lambda = \dfrac {1 + \sqrt 5}2 = \varphi \approx \boxed{1.618} \end{aligned}$