A particularly interesting sequence of natural numbers is called after its early discoverer Leonardo Fibonacci (ca. 1200).

Fibonacci It is created by defining each terms as the sum of its two predecessors. Thus the formation law reads:

$$\begin{array}{c}{A}_{0}=0;{A}_{1}=1\hfill \\ {A}_{n+2}={A}_{n}+{A}_{n+1}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}for\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}n\ge 0\hfill \\ \hfill \end{array}$$

The first 25 numbers in the sequence are:

0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610; 987; 1597; 2584; 4181; 6765; 10946; 17711; 28657; 46368; 75025

The ratio ${A}_{n}\u2215{A}_{n-1}$ of consecutive terms converges very quickly to the irrational value of the golden mean (In art the golden mean is a criterion for the balance of proportions: two dimension adhere to the golden mean, if ratio of the larger one to the smaller one is the same as the ration of the sum of both to the larger one ).

$${A}_{n}\u2215{A}_{n-1}\to \Phi =1.618033988.....$$

The first values, which can be easily obtained with an Excel spread sheet are

1.0; 2.0; 1.5; 1.6666666667; 1.6; 1.625; 1.6153846154; 1.6190476190; 1.6176470588; 1.6181818182; 1.6179775281; 1.6180555556; 1.6180257511; 1.6180371353; 1.6180327869; 1.6180344478; 1.6180338134; 1.6180340557; 1.6180339632

It is evident, that the differences of consecutive terms to $\Phi $ alternate in sign. In this sign the approximation to the golden mean happens in oscillating manner

This ratio can also be represented as continued fraction with $n-1$ fractions (Please try this out for the first few terms) :

$${A}_{n}\u2215{A}_{n-1}=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+.......}}}}}}\to \Phi $$

From this one easily obtains, that $\Phi =\frac{2}{\sqrt{5}-1}$ as positive root of the equation

$$\Phi =1+\frac{1}{\Phi}\to {\Phi}^{2}-\Phi -1=0\phantom{\rule{0em}{0ex}}.$$

For the exponential sequence we have from the first term onwards:

$${A}_{n}\u2215{A}_{n-1}=\frac{{e}^{n}}{{e}^{n-1}}=e=2.718...$$

While the sequence of ratios is constant from the beginning for the exponential sequence, the ratios for the Fibonacci sequence only approximate a constant value for $n\to \infty $ . For large $n$ both sequences are obviously similar. From this analogy one can deduce, that the Fibonacci sequence approximates an exponential sequence for $n\to \infty $. This is sign, that the Fibonacci sequence can describe growth processes just as the exponential function.

There exist many arithmetic relationships between the terms of the Fibonacci sequence. In addition there are many interesting application to problems of symmetry and growth, for which we refer to the given links.