Fibonacci Sequence Calculator

The Fibonacci sequence is a series of numbers where each term equals the sum of the two preceding terms. It starts with F(0) = 0 and F(1) = 1, then continues: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Each number is defined by the rule F(n) = F(n-1) + F(n-2).

You can calculate the nth Fibonacci number iteratively by summing the two previous terms, or directly using Binet's formula: F(n) = ((1+√5)^n − (1−√5)^n) / (2^n × √5). This calculator uses Binet's formula for single-term lookups and iterative generation for sequences.

The first 10 Fibonacci numbers (starting from F(0)) are: 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. If you start from F(1), they are: 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.

The golden ratio (approximately 1.6180339887) is the limit of the ratio of consecutive Fibonacci numbers — F(n)/F(n-1) — as n approaches infinity. This ratio appears throughout nature, art, and architecture and is often denoted by the Greek letter phi (φ).

Binet's formula is a closed-form expression: F(n) = (φ^n − ψ^n) / √5, where φ = (1+√5)/2 ≈ 1.618 (the golden ratio) and ψ = (1−√5)/2 ≈ −0.618. It allows you to compute any Fibonacci number directly without calculating all previous terms.

Yes — the Fibonacci sequence can be extended into negative indices using the rule F(−n) = (−1)^(n+1) × F(n). For example, F(−1) = 1, F(−2) = −1, F(−3) = 2. This is sometimes called the negafibonacci sequence.

Fibonacci numbers appear in many fields: in nature (flower petal counts, spiral patterns in shells and galaxies), in computer science (search algorithms and data structures), in financial trading (Fibonacci retracement levels), and in art and architecture for proportional design.

Fibonacci numbers grow exponentially — they roughly multiply by the golden ratio (~1.618) with each step. F(100) already exceeds 354 billion billion, and F(500) is a number with over 100 digits. This calculator accurately computes terms up to n = 100 before precision limits of floating-point arithmetic become significant.