Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less
Jump to navigation
Jump to search
Theorem
Let $n \in \Z$.
Then:
- $\hat \phi^n = F_n \hat \phi + F_{n - 1}$
where:
- $F_n$ denotes the $n$th Fibonacci number
- $\hat \phi$ denotes the $1$ minus the golden mean:
- $\hat \phi := 1 - \phi$
Proof
\(\ds F_n \hat \phi + F_{n - 1}\) | \(=\) | \(\ds F_n \left({-\dfrac 1 \phi}\right) + F_{n - 1}\) | Reciprocal Form of One Minus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 \phi \left({F_n - \phi F_{n - 1} }\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 \phi \left({\left({-1}\right)^{n + 1} F_{-n} - \phi \left({-1}\right)^n F_{-\left({n - 1}\right)} }\right)\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 \phi \left({\left({-1}\right)^{n + 1} F_{-n} + \phi \left({-1}\right)^{n + 1} F_{-\left({n - 1}\right)} }\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({-1}\right)^{n + 1} \left({-\dfrac 1 \phi \left({F_{-n} + \phi F_{-n + 1} }\right)}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({-1}\right)^{n + 1} \left({-\dfrac 1 \phi }\right) \left({\phi^{-n + 1} }\right)\) | Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less | |||||||||||
\(\ds \) | \(=\) | \(\ds \left({-1}\right)^n \left({\dfrac 1 {\phi^n} }\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({-\dfrac 1 \phi}\right)^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \hat \phi^n\) |
$\blacksquare$
Also see
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 $11$