Fibonacci Sequence

The Fibonacci sequence is a sequence of numbers in which each number after the first two is the sum of the two preceding ones. This sequence was introduced to the western world in the 13th century by the Italian mathematician Leonardo of Pisa, also known as Fibonacci, though it was described in Indian mathematics much earlier.

Definition

The Fibonacci sequence is defined by the following recurrence relation:

F(0) = 0, F(1) = 1
F(n) = F(n-1) + F(n-2) for n > 1

In other words, each number is the sum of the two preceding ones, starting from 0 and 1.

The first few numbers in the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Properties of the Fibonacci Sequence

The Fibonacci sequence has several interesting properties:

  1. Golden Ratio: The ratio of successive Fibonacci numbers converges to the golden ratio (approximately 1.61803) as n goes to infinity.

  2. Cassini's Identity: For any positive integer n, F(n)² - F(n-1) * F(n+1) = (-1)^(n+1).

  3. Periodicity Modulo m: For any positive integer m, the sequence F(n) mod m is periodic.

Applications

The Fibonacci sequence appears in many different areas of mathematics and science. It is used in the computational run-time study of Euclid's algorithm, it appears in solutions to certain differential equations, and it even emerges in certain counting problems. In addition, the Fibonacci sequence is seen in crystallographic structures and in phyllotaxis (the arrangement of leaves on a stem in some plants). It also has applications in computer algorithms and programming.