Fibonacci Number n+1 Minus Golden Mean by Fibonacci Number n
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Theorem
- $F_{n + 1} - \phi F_n = \hat \phi^n$
where:
- $F_n$ denotes the $n$th Fibonacci number
- $\phi$ denotes the golden mean.
Proof 1
\(\ds F_{n + 1} - \phi F_n\) | \(=\) | \(\ds \dfrac 1 {\sqrt 5} \paren {\phi^{n + 1} - \hat \phi^{n + 1} } - \dfrac \phi {\sqrt 5} \paren {\phi^n - \hat \phi^n}\) | Euler-Binet Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 5} \paren {\phi^{n + 1} - \hat \phi^{n + 1} - \phi^{n + 1} + \phi \hat \phi^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \hat \phi^n \dfrac {\phi - \hat \phi} {\sqrt 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \hat \phi^n F_1\) | Euler-Binet Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \hat \phi^n\) | Definition of Fibonacci Number: $F_1 = 1$ |
$\blacksquare$
Proof 2
\(\ds F_n\) | \(=\) | \(\ds \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}\) | Euler-Binet Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^n - \hat \phi^n} {\frac {1 + \sqrt 5} 2 - \frac {1 - \sqrt 5} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\phi^n - \hat \phi^n} {\phi - \hat \phi}\) | Definition 2 of Golden Mean |
Thus from Recurrence Relation where n+1th Term is A by nth term + B to the n we have:
- $F_{n + 1} = \phi F_n + \hat \phi^n$
whence the result.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $28$