Number Theory
Advanced

Binet Formula

Closed form for the nth Fibonacci number.

Formula

Fn=φn−ψn5F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}

Variables

\varphiGolden ratio
\psi1-\varphi

Example

Direct Fibonacci without recursion

Did You Know?

Binet's formula computes any Fibonacci number directly using the golden ratio, no counting required.

sequence