Fibonacci sequence

Fibonacci Squares

Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. That is, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. The sequence bears the name of Leonardo of Pisa, known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci, although the sequence had been previously described in Indian mathematics.

Definition

The Fibonacci sequence is defined by the recurrence relation:

F(n) = F(n-1) + F(n-2)

with seed values:

F(0) = 0, F(1) = 1

This means that each term is the sum of the previous two terms, starting from 0 and 1.

Properties

The Fibonacci sequence exhibits a vast array of properties and appears in various aspects of art, architecture, music, and nature. Notably, the ratio of successive Fibonacci numbers approximates the Golden Ratio (approximately 1.618033988749895...), as the numbers increase. This ratio is often symbolized by the Greek letter φ (phi) and is known for its aesthetically pleasing properties.

Applications

The Fibonacci sequence is applied in numerous fields, including computational algorithms, financial markets, and the modeling of biological systems. In computer science, Fibonacci numbers are used in algorithms for sorting and searching. In finance, the Fibonacci retracement technique is used to predict future movements of financial assets. In nature, Fibonacci numbers are observable in the arrangement of leaves on a stem, the branching of trees, the flowering of artichoke, the uncurling of ferns, and the arrangement of a pine cone's bracts.

Mathematical Relationships

Several interesting mathematical relationships involve the Fibonacci sequence. For example, the sum of the first n Fibonacci numbers is equal to the (n + 2)nd Fibonacci number minus 1. Additionally, the squares of the Fibonacci numbers, and the products of adjacent Fibonacci numbers, have their own unique properties and relationships.

In Popular Culture

The Fibonacci sequence has also captured the public imagination and appears in various aspects of popular culture, including literature, movies, and puzzles. It is often presented as an example of the inherent beauty and order in mathematics and nature.

See Also

• Golden Ratio
• Leonardo of Pisa
• Liber Abaci
• Mathematical beauty
• Patterns in nature