What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. Formally, the sequence is defined by the recurrence relation: **F(n) = F(n-1) + F(n-2)** with initial values: **F(0) = 0, F(1) = 1.** This generates the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Mathematically, the Fibonacci sequence has many interesting properties. For example, the ratio of consecutive Fibonacci numbers approximates the golden ratio (approximately 1.618), which appears frequently in nature, art, and architecture. The sequence also has connections to combinatorics, number theory, and even the solution of certain types of equations. In practical terms, the Fibonacci sequence is used in computer algorithms, such as sorting and searching, and appears in modeling population growth, financial market analysis, and biological settings like the arrangement of leaves or the branching patterns of trees. Its recursive nature makes it a classic example when teaching programming and mathematical induction.