What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. Formally, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Mathematically, it can be defined by the recurrence relation: **F(n) = F(n-1) + F(n-2)**, with initial values **F(0) = 0** and **F(1) = 1**. What makes the Fibonacci sequence special is both its simplicity and its surprising presence in nature, mathematics, and art. For example, the sequence appears in the arrangement of leaves on a stem, the branching of trees, the pattern of seeds in a sunflower, and the spirals of shells. This connection to natural growth patterns is linked to the way the sequence approximates the golden ratio (approximately 1.618), which is often associated with aesthetically pleasing proportions. In mathematics, the Fibonacci sequence has many interesting properties and applications. It appears in number theory, computer algorithms, and even in financial markets for technical analysis. Its recursive nature also makes it a popular example in teaching programming and algorithm design. Overall, the Fibonacci sequence is a fascinating intersection of nature, mathematics, and art.