Fibonaccinumber

Fibonaccinumber, more commonly known as the Fibonacci number, refers to any term F(n) of the Fibonacci sequence. The sequence is defined by the recurrence F(n) = F(n−1) + F(n−2) for n ≥ 2, with initial values F(0) = 0 and F(1) = 1. Subsequent terms begin 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Some sources use a different indexing, such as F(1) = 1 and F(2) = 1.

A closed-form expression for F(n) is given by Binet’s formula: F(n) = (phi^n − psi^n)/√5, where phi = (1 + √5)/2 is the golden ratio and psi = (1 − √5)/2. Because |psi| < 1, the second term becomes negligible for large n, making F(n) grow approximately like phi^n / √5.

Fibonacci numbers possess several notable properties. They are integers for all n and exhibit a gcd relation: gcd(F(m), F(n)) = F(gcd(m, n)). They also appear in connection with the golden ratio, since F(n) ≈ phi^n / √5 as n grows. In modular arithmetic, Fibonacci numbers are periodic modulo any positive integer, a phenomenon known as the Pisano period.

Applications of Fibonacci numbers appear across mathematics, computer science, and nature. They arise in algorithmic design (fast doubling methods for efficient computation), counting problems, and tiling. They also appear in biological settings and pattern formation, where ratios and growth sequences resemble Fibonacci progressions. Related sequences include the Lucas numbers, which satisfy a similar recurrence with different initial conditions.