Let $u_n$ be the $n$-th entry in the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$

Question

If you start with $u_1 = 1$ and $u_2 = 1$, then the sequence can be generated using the formula $$u_{n+1} = u_n + u_{n-1}\ .$$ If $u_n = r^n$, what is r?

Can anyone figure this out? I am so stuck please help!

Answer 1

$F_n$ is approximately equal to $\dfrac{\Phi^n}{\sqrt{5}}$, where $\Phi$ is the golden ratio, $\Phi = \dfrac{\sqrt{5}+1}{2}$.

If $F_{n+2} = F_{n+1} + F_n$ and $F_{n+1} = rF_n$, then $r^2F_n = rF_n + F_n$ and $r^2 = r + 1$ or $r^2 - r - 1 = 0$. This quadratic has roots $\Phi$ and $1 - \Phi = \dfrac{1 - \sqrt{5}}{2}$ which is negative.