Two large Fibonacci numbers

Number Theory Level 3

Consider the Fibonacci numbers :

$F_0 = 0$
$F_1 = 1$
$F_n = F_{n-1} + F_{n-2}$ for $n > 1$

$\ln (F_{8046}) - \ln (F_{8045}) = \text{ } ?$

(Provide your answer to three decimal places)

The answer is 0.481.

1 solution

Jordan Cahn
Jan 14, 2019

$\begin{aligned} \ln(F_{8046}) - \ln(F_{8045}) &= \ln\left(\frac{F_{8046}}{F_{8045}}\right) \\ &\approx \ln(\phi) && \color{#3D99F6} \frac{F_{n+1}}{F_n} \to \phi \text{ as } n\to\infty\text{, where }\phi\text{ is the Golden Ratio.} \\ &\approx \boxed{0.4812} \end{aligned}$

Two large Fibonacci numbers

The answer is 0.481.

1 solution

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