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.