Fibonacci sequence

The Fibonacci sequence is a sequence of integers, starting from 0 and 1, such that the sum of the preceding two integers is the following number in the sequence. The numbers in this sequence are referred to as Fibonacci numbers. Mathematically, for n>1, the Fibonacci sequence can be described as follows:

F0 = 0

F1 = 1

Fn = Fn-1 + Fn-2

The beginning of the sequence is thus:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

As can be seen from the above sequence, and using the above notation,

F2 = F1 + F0 = 1 + 0 = 1

F3 = F2 + F1 = 1 + 1 = 2

F4 = F3 + F2 = 2 + 1 = 3


and so on.

Fibonacci numbers are strongly related to the golden ratio. The bigger the pair of Fibonacci numbers used, the closer their ratio is to the golden ratio. Fibonacci numbers are seen often enough in math, as well as nature, that they are a subject of study. They are used in certain computer algorithms, can be seen in the branching of trees, arrangement of leaves on a stem, and more.

The golden ratio

The golden ratio, often represented using the Greek letter phi (Φ), is an irrational number:

Two numbers exhibit the golden ratio if the ratio of the two numbers is equal to the ratio of the sum of the two numbers to that of the larger number. Mathematically, given two positive numbers, a and b, where a is the larger number, this can be written as: