Euler-Binet Formula

Theorem
The Fibonacci numbers have a closed-form solution:
 * $\displaystyle F \left({n}\right) = \frac {\phi^n - \left({1 - \phi}\right)^n} {\sqrt 5} = \frac {\phi^n - \left({-1 / \phi}\right)^n} {\sqrt 5} = \frac {\phi^n - \left({-1}\right)^n\phi^{-n} } {\sqrt 5}$

where $\phi$ is the golden mean.

Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:
 * $\displaystyle F \left({n}\right) = \frac {\phi^n - \hat \phi^n} {\sqrt 5}$

Proof 3
It is also known as Binet's Formula.

Binet derived it in 1843, but it was already known to Euler, de Moivre and Daniel Bernoulli over a century earlier.

However, it was Binet who derived the more general Binet Form of which this is an elementary application.