What is the Fibonacci sequence?

The Fibonacci sequence is a famous series of numbers in mathematics where each number is the sum of the two preceding ones. It usually starts with 0 and 1, so the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Formally, it can be defined by the recurrence relation: \[ F_0 = 0, \quad F_1 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 2 \] This simple rule generates a sequence with many interesting properties and connections. The Fibonacci sequence appears in many areas of mathematics, including number theory, combinatorics, and algebra. For example, it shows up in the study of the golden ratio, since the ratio of consecutive Fibonacci numbers \( F_{n+1} / F_n \) approaches the golden ratio (approximately 1.618) as \( n \) becomes large. It also appears in counting problems, such as the number of ways to climb stairs taking one or two steps at a time, and in the branching patterns of trees and plants in nature. Beyond pure math, Fibonacci numbers are often found in biology, art, and architecture, illustrating how mathematics can describe natural patterns and aesthetic proportions.