# Euler-Binet Formula/Proof 3

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## Theorem

The Fibonacci numbers have a closed-form solution:

- $F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5}$

where $\phi$ is the golden mean.

Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:

- $F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$

## Proof

This follows as a direct application of the first Binet form:

- $U_n = m U_{n - 1} + U_{n - 2}$

where:

\(\ds U_0\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds U_1\) | \(=\) | \(\ds 1\) |

has the closed-form solution:

- $U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$

where:

\(\ds \Delta\) | \(=\) | \(\ds \sqrt {m^2 + 4}\) | ||||||||||||

\(\ds \alpha\) | \(=\) | \(\ds \frac {m + \Delta} 2\) | ||||||||||||

\(\ds \beta\) | \(=\) | \(\ds \frac {m - \Delta} 2\) |

where $m = 1$.

$\blacksquare$

## Source of Name

This entry was named for Jacques Philippe Marie Binet and Leonhard Paul Euler.

## Also known as

The **Euler-Binet Formula** is also known as **Binet's formula**.