Ratio of Consecutive Lucas Numbers

Theorem

For $n \in \N$, let $L_n$ be the $n$th Lucas number.

Then:

$\ds \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n} = \phi$

where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.

$\ds \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n}$

$=$

$\ds \lim_{n \mathop \to \infty} \frac {\phi^{n + 1} + \paren {-\phi^{-1} }^{n + 1} } {\phi^n + \paren {-\phi^{-1} }^n}$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \frac {\phi - \phi^{-1} \paren {-\phi^{-2} }^n } {1 + \paren {-\phi^{-2} }^n}$

dividing both numerator and denominator by $\phi^n$

$\ds $

$=$

$\ds \frac \phi 1$

$\ds $

$=$

$\ds \phi$

$\blacksquare$