Upper and Lower Bound of Fibonacci Number

Theorem

For all $n \in \N_{> 0}$:

$\phi^{n - 2} \le F_n \le \phi^{n - 1}$

where:

$F_n$ is the $n$th Fibonacci number

$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$

Proof

From Fibonacci Number greater than Golden Section to Power less Two:

$F_n \ge \phi^{n - 2}$

From Fibonacci Number less than Golden Section to Power less One:

$F_n \le \phi^{n - 1}$

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers