Power of Golden Mean as Sum of Smaller Powers
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Theorem
Let $\phi$ denote the golden mean.
Then:
- $\forall z \in \C: \phi^z = \phi^{z - 1} + \phi^{z - 2}$
Proof
Let $z \in \C$.
Let $w \in \C$ such that $w + 2 = z$.
Then:
\(\ds \phi^z\) | \(=\) | \(\ds \phi^{w + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^w \phi^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^w \left({\phi + 1}\right)\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{w + 1} + \phi^w\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{z - 1} + \phi^{z - 2}\) |
$\blacksquare$
Sources
- 1957: George Bergman: Number System with an Irrational Base (Math. Mag. Vol. 31, no. 2: pp. 98 – 110) www.jstor.org/stable/3029218