Binet's formula

Builds on Fibonacci sequence Golden ratio

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 $\varphi^n / \sqrt{5}$ .

Reading path

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Tag	`0404D8TN7Y`
URI	`m2://0404D8TN7Y`
Kind	statement, theorem
Writing	W1 stated
MSC	11B39
Status	This copy of the site has no build data. Run `m2 status 0404D8TN7Y`.
Uses	Fibonacci sequence, Golden ratio

Binet's formula

Statement

Remarks

Reading path

Further reading

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