Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less

Theorem
Let $n \in \Z$.

Then:


 * $\phi^n = F_n \phi + F_{n - 1}$

where:
 * $F_n$ denotes the $n$th Fibonacci number
 * $\phi$ denotes the golden mean.

Positive Index
First the result is proved for positive integers.

Negative Index
Then the result is extended to negative integers.

Hence the result is seen to show for both positive and negative integers.

Also see

 * Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less