Binet's formula

Statement

Let $n$ be a natural number, and let $F_n$ be the $n$-th Fibonacci number. Let $\varphi = \frac{1 + \sqrt{5}}{2}$ be the golden ratio and let $\psi = \frac{1 - \sqrt{5}}{2}$. Then $$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}.$$

Remarks

The proof uses the characteristic polynomial $x^2 - x - 1$ of the recurrence. Its roots are $\varphi$ and $\psi$, so $\varphi^n$ and $\psi^n$ are solutions, and the initial values fix the coefficients. Because $|\psi^n / \sqrt{5}| < \frac{1}{2}$, $F_n$ is the integer nearest to $\varphi^n / \sqrt{5}$.

Explanations

L1 human

Fibonacci numbers are integers, but this formula uses $\sqrt{5}$, and the irrational parts cancel. The term with $\psi^n$ is very small, so $F_n$ is close to $\varphi^n / \sqrt{5}$.