Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Positive Index

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Theorem

Let $n \in \Z_{\ge 0}$.

Then:

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

where:

$F_n$ denotes the $n$th Fibonacci number
$\phi$ denotes the golden mean.


Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:

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


$P \left({0}\right)$ is the case:

\(\ds F_0 \times \phi + F_{-1}\) \(=\) \(\ds F_0 \times \phi + \left({-1}\right)^0 F_1\) Fibonacci Number with Negative Index
\(\ds \) \(=\) \(\ds F_0 \times \phi + 1\) Definition of Fibonacci Number $F_1 = 1$
\(\ds \) \(=\) \(\ds 0 \times \phi + 1\) Definition of Fibonacci Number $F_0 = 0$
\(\ds \) \(=\) \(\ds 1\)
\(\ds \) \(=\) \(\ds \phi^0\)


Thus $P \left({0}\right)$ is seen to hold.


Basis for the Induction

$P \left({1}\right)$ is the case:

\(\ds F_1 \times \phi + F_0\) \(=\) \(\ds F_1 \times \phi\) Definition of Fibonacci Number $F_0 = 0$
\(\ds \) \(=\) \(\ds 1 \times \phi\) Definition of Fibonacci Number $F_1 = 1$
\(\ds \) \(=\) \(\ds \phi^1\)


Thus $P \left({1}\right)$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k + 1}\right)$ is true.


So this is the induction hypothesis:

$\phi^k = F_k \phi + F_{k - 1}$


from which it is to be shown that:

$\phi^{k + 1} = F_{k + 1} \phi + F_k$


Induction Step

This is the induction step:


\(\ds F_{k + 1} \phi + F_k\) \(=\) \(\ds \left({F_k + F_{k - 1} }\right) \phi + F_k\) Definition of Fibonacci Number
\(\ds \) \(=\) \(\ds F_k \left({1 + \phi}\right) + F_{k - 1} \phi\)
\(\ds \) \(=\) \(\ds F_k \phi^2 + F_{k - 1} \phi\) Square of Golden Mean equals One plus Golden Mean
\(\ds \) \(=\) \(\ds \phi \left({F_k \phi + F_{k - 1} }\right)\)
\(\ds \) \(=\) \(\ds \phi \left({\phi^n}\right)\) Induction Hypothesis
\(\ds \) \(=\) \(\ds \phi^{n + 1}\)


So $P \left({k}\right) \implies P \left({k + 1}\right)$ and it follows by the Principle of Mathematical Induction that:

$\forall n \in \Z_{\ge 0}: \phi^n = F_n \phi + F_{n - 1}$

$\blacksquare$