fibonacci

Fibonacci Sequence

The Fibonacci numbers $F_n$ are defined recursively:

$$F_n=\{\begin{array}{ll}0& \text{if }n=0\\ 1& \text{if }n=1\\ F_{n-1}+F_{n-2}& \text{if }n>1\end{array}$$

Generating the sequence: $0,1,1,2,3,5,8,13,21,34,55,89,144,...$

Each number is the sum of the previous two.
This recursive structure appears throughout nature (spiral patterns in shells, seed arrangements in sunflowers) and has deep mathematical connections.

Closed-Form Solution (Binet's Formula) – Fibonacci numbers grow exponentially with base $\varphi$.

$$F_n=\frac{{\varphi }^n-{\psi }^n}{\sqrt{5}}$$

where $\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618$ (golden ratio) and $\psi =\frac{1-\sqrt{5}}{2}\approx -0.618$
See also: Eigenvector puzzle.

Since $|\psi |<1$, the term ${\psi }^n$ vanishes as $n$ grows, so for large $n$:

$$F_n\approx \frac{{\varphi }^n}{\sqrt{5}}$$

The golden ratio $\varphi$ is the smallest possible base for which a sequence satisfying $F_n=F_{n-1}+F_{n-2}$ can grow exponentially . Any slower growth would eventually turn negative or decay to zero.

Golden Ratio Convergence

The ratio of consecutive Fibonacci numbers converges to the golden ratio:

$$\underset{n\to \infty }{\lim }\frac{F_{n+1}}{F_n}=\varphi$$

This follows from the recurrence relation: if $\frac{F_{n+1}}{F_n}\to L$, then $\frac{F_{n+2}}{F_{n+1}}=\frac{F_{n+1}+F_n}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}\to 1+\frac{1}{L}$.
At the limit, $L=1+\frac{1}{L}$, giving $L^2=L+1$, whose positive solution is $\varphi =\frac{1+\sqrt{5}}{2}$.

$\frac{1}{1}=1$, $\frac{2}{1}=2$, $\frac{3}{2}=1.5$, $\frac{5}{3}\approx 1.667$, $\frac{8}{5}=1.6$, $\frac{13}{8}=1.625$, $\frac{21}{13}\approx 1.615$, … → $\varphi \approx 1.618$