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