# Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less/Negative Index

## Theorem

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

Then:

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

where:

$F_n$ denotes the $n$th Fibonacci number as extended to negative indices
$\phi$ denotes the golden mean.

## Proof

The proof proceeds by induction.

For all $n \in \Z_{\le 0}$, let $\map P n$ be the proposition:

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

This can equivalently be expressed as:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

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

$\map P 0$ is the case:

 $\ds F_0 \times \phi + F_{-1}$ $=$ $\ds F_0 \times \phi + \paren {-1}^0 F_1$ 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 $\map P 0$ is seen to hold.

#### Basis for the Induction

$\map P 1$ is the case:

 $\ds F_{-1} \times \phi + F_{-2}$ $=$ $\ds \paren {-1}^0 F_1 \times \phi + \paren {-1}^{-1} F_2$ Fibonacci Number with Negative Index $\ds$ $=$ $\ds 1 \times \phi - F_2$ Definition of Fibonacci Number $F_1 = 1$ $\ds$ $=$ $\ds \phi - 1$ Definition of Fibonacci Number: $F_2 = 1$ $\ds$ $=$ $\ds \dfrac 1 \phi$ Definition 3 of Golden Mean

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

#### Induction Hypothesis

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

So this is the induction hypothesis:

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

from which it is to be shown that:

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

#### Induction Step

This is the induction step:

 $\ds F_{-\paren {k + 1} } \phi + F_{-\paren {k + 1} - 1}$ $=$ $\ds \paren {-1}^{-\paren {k + 1} + 1} F_{k + 1} \phi + \paren {-1}^{-\paren {k + 1} } F_{\paren {k + 1} + 1}$ $\ds$ $=$ $\ds \paren {-1}^{k + 2} F_{k + 1} \phi - \paren {-1}^{k + 2} F_{k + 2}$ $\ds$ $=$ $\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \phi - \paren {F_{k + 1} + F_k} }$ Definition of Fibonacci Number $\ds$ $=$ $\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \paren {\phi - 1} - F_k}$ $\ds$ $=$ $\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \phi^{-1} - F_k}$ Definition 3 of Golden Mean $\ds$ $=$ $\ds \paren {-1}^{k + 2} F_{k + 1} \phi^{-1} + \paren {-1}^{k + 1} F_k$ $\ds$ $=$ $\ds \frac 1 \phi \paren {F_{-k - 1} + F_{-k} \phi}$ Fibonacci Number with Negative Index $\ds$ $=$ $\ds \frac 1 \phi \paren {\phi^{-k} }$ Induction Hypothesis $\ds$ $=$ $\ds \phi^{-\paren {k + 1} }$

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

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

$\blacksquare$