What is Fibonacci Sequence of numbers?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It typically starts with 0 and 1. The sequence begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on[1][2][3][4][5][6][7][8].

Definition and Formula

Mathematically, the Fibonacci sequence can be defined by the recurrence relation:
$$ F_{n} = F_{n-1} + F_{n-2} $$
with initial conditions:
$$ F_{0} = 0 $$
$$ F_{1} = 1 $$

Historical Context

The sequence is named after Leonardo of Pisa, known as Fibonacci, who introduced it to Western mathematics in his 1202 book "Liber Abaci"[5][6]. However, the sequence was previously described in Indian mathematics as early as 200 BC in work by Pingala[5].

Properties and Applications

• Golden Ratio: The ratio of successive Fibonacci numbers approximates the golden ratio (approximately 1.6180339887), which is significant in various fields such as art, architecture, and nature[1][3][4][8].
• Natural Occurrences: Fibonacci numbers frequently appear in biological settings, such as the arrangement of leaves on a stem, the fruit sprouts of a pineapple, and the spiral patterns of shells and galaxies[3][...