Fibonacci Number less than Golden Section to Power less One
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Theorem
For all $n \in \N_{> 0}$:
- $F_n \le \phi^{n - 1}$
where:
- $F_n$ is the $n$th Fibonacci number
- $\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$
Proof
The proof proceeds by induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
- $F_n \le \phi^{n - 1}$
Basis for the Induction
$\map P 1$ is true, as this just says:
- $F_1 = 1 = \phi^0 = \phi^{1 - 1}$
It is also necessary to demonstrate $\map P 2$ is true:
- $F_2 = 1 \le \dfrac {1 + \sqrt 5} 2 = \phi = \phi^{2 - 1}$
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, for all $1 \le k \le n$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $F_k \le \phi^{k - 1}$
from which it is to be shown that:
- $F_{k + 1} \le \phi^k$
Induction Step
This is the induction step:
\(\ds F_{k + 1}\) | \(=\) | \(\ds F_{k - 1} + F_k\) | Definition of Fibonacci Numbers | |||||||||||
\(\ds \) | \(\le\) | \(\ds \phi^{k - 2} + \phi^{k - 1}\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{k - 2} \paren {1 + \phi}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{k - 2} \paren {\phi^2}\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^k\) |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N_{> 0}: F_n \le \phi^{n - 1}$
$\blacksquare$
Also see
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: $(3)$