Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less
Theorem
Let $n \in \Z$.
Then:
- $\phi^n = F_n \phi + F_{n - 1}$
where:
- $F_n$ denotes the $n$th Fibonacci number
- $\phi$ denotes the golden mean.
Proof
Positive Index
First the result is proved for positive integers.
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
- $\phi^n = F_n \phi + F_{n - 1}$
$P \left({0}\right)$ is the case:
\(\ds F_0 \times \phi + F_{-1}\) | \(=\) | \(\ds F_0 \times \phi + \left({-1}\right)^0 F_1\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \) | \(=\) | \(\ds F_0 \times \phi + 1\) | Definition of Fibonacci Number $F_1 = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 \times \phi + 1\) | Definition of Fibonacci Number $F_0 = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^0\) |
Thus $P \left({0}\right)$ is seen to hold.
Basis for the Induction
$P \left({1}\right)$ is the case:
\(\ds F_1 \times \phi + F_0\) | \(=\) | \(\ds F_1 \times \phi\) | Definition of Fibonacci Number $F_0 = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times \phi\) | Definition of Fibonacci Number $F_1 = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^1\) |
Thus $P \left({1}\right)$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k + 1}\right)$ is true.
So this is the induction hypothesis:
- $\phi^k = F_k \phi + F_{k - 1}$
from which it is to be shown that:
- $\phi^{k + 1} = F_{k + 1} \phi + F_k$
Induction Step
This is the induction step:
\(\ds F_{k + 1} \phi + F_k\) | \(=\) | \(\ds \left({F_k + F_{k - 1} }\right) \phi + F_k\) | Definition of Fibonacci Number | |||||||||||
\(\ds \) | \(=\) | \(\ds F_k \left({1 + \phi}\right) + F_{k - 1} \phi\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_k \phi^2 + F_{k - 1} \phi\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi \left({F_k \phi + F_{k - 1} }\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi \left({\phi^n}\right)\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{n + 1}\) |
So $P \left({k}\right) \implies P \left({k + 1}\right)$ and it follows by the Principle of Mathematical Induction that:
- $\forall n \in \Z_{\ge 0}: \phi^n = F_n \phi + F_{n - 1}$
$\Box$
Negative Index
Then the result is extended to negative integers.
The proof proceeds by induction.
For all $n \in \Z_{\le 0}$, let $\map P n$ be the proposition:
- $\phi^n = F_n \phi + F_{n - 1}$
This can equivalently be expressed as:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $\phi^{-n} = F_{-n} \phi + F_{-n - 1}$
$\map P 0$ is the case:
\(\ds F_0 \times \phi + F_{-1}\) | \(=\) | \(\ds F_0 \times \phi + \paren {-1}^0 F_1\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \) | \(=\) | \(\ds F_0 \times \phi + 1\) | Definition of Fibonacci Number $F_1 = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 \times \phi + 1\) | Definition of Fibonacci Number $F_0 = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^0\) |
Thus $\map P 0$ is seen to hold.
Basis for the Induction
$\map P 1$ is the case:
\(\ds F_{-1} \times \phi + F_{-2}\) | \(=\) | \(\ds \paren {-1}^0 F_1 \times \phi + \paren {-1}^{-1} F_2\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times \phi - F_2\) | Definition of Fibonacci Number $F_1 = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi - 1\) | Definition of Fibonacci Number: $F_2 = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 \phi\) | Definition 3 of Golden Mean |
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\phi^{-k} = F_{-k} \phi + F_{-k - 1}$
from which it is to be shown that:
- $\phi^{-\paren {k + 1} } = F_{-\paren {k + 1} } \phi + F_{-\paren {k + 1} - 1}$
Induction Step
This is the induction step:
\(\ds F_{-\paren {k + 1} } \phi + F_{-\paren {k + 1} - 1}\) | \(=\) | \(\ds \paren {-1}^{-\paren {k + 1} + 1} F_{k + 1} \phi + \paren {-1}^{-\paren {k + 1} } F_{\paren {k + 1} + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{k + 2} F_{k + 1} \phi - \paren {-1}^{k + 2} F_{k + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \phi - \paren {F_{k + 1} + F_k} }\) | Definition of Fibonacci Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \paren {\phi - 1} - F_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{k + 2} \paren {F_{k + 1} \phi^{-1} - F_k}\) | Definition 3 of Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{k + 2} F_{k + 1} \phi^{-1} + \paren {-1}^{k + 1} F_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \phi \paren {F_{-k - 1} + F_{-k} \phi}\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \phi \paren {\phi^{-k} }\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-\paren {k + 1} }\) |
So $\map P k \implies \map P {k + 1}$ and it follows by the Principle of Mathematical Induction that:
- $\forall n \in \Z_{\le 0}: \phi^n = F_n \phi + F_{n - 1}$
$\Box$
Hence the result is seen to show for both positive and negative integers.
$\blacksquare$
Also see
Sources
- 1957: George Bergman: Number System with an Irrational Base (Math. Mag. Vol. 31, no. 2: pp. 98 – 110) www.jstor.org/stable/3029218
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $11$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$