Fibonacci Number n+1 Minus Golden Mean by Fibonacci Number n/Proof 1

Theorem

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

where:

$F_n$ denotes the $n$th Fibonacci number

$\phi$ denotes the golden mean.

$\ds F_{n + 1} - \phi F_n$

$=$

$\ds \dfrac 1 {\sqrt 5} \paren {\phi^{n + 1} - \hat \phi^{n + 1} } - \dfrac \phi {\sqrt 5} \paren {\phi^n - \hat \phi^n}$

$\ds $

$=$

$\ds \dfrac 1 {\sqrt 5} \paren {\phi^{n + 1} - \hat \phi^{n + 1} - \phi^{n + 1} + \phi \hat \phi^n}$

$\ds $

$=$

$\ds \hat \phi^n \dfrac {\phi - \hat \phi} {\sqrt 5}$

$\ds $

$=$

$\ds \hat \phi^n F_1$

$\ds $

$=$

$\ds \hat \phi^n$

Definition of Fibonacci Number: $F_1 = 1$

$\blacksquare$