Euler-Binet Formula/Proof 3

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

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


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

where:

has the closed-form solution:
 * $U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$

where:

where $m=1$.

It is also known as Binet's Formula.