Fibonacci Calculator

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. Starting with F(1) = 1 and F(2) = 1, the sequence continues 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. The formula is F(n) = F(n-1) + F(n-2) for n > 2.

As you move further along the Fibonacci sequence, the ratio of consecutive numbers approaches the golden ratio, approximately 1.618033989. More precisely, F(n) / F(n-1) converges to phi (φ), which equals (1 + sqrt(5)) / 2. This special number appears throughout nature, art, and architecture.

Yes. The most common convention starts with F(1) = 1, F(2) = 1. However, some definitions begin with F(0) = 0, F(1) = 1, giving the sequence 0, 1, 1, 2, 3, 5, 8, and so on. This calculator shows both conventions so you can use the one that matches your needs.

Fibonacci patterns appear in spiral flower seed heads, the branching of trees, the arrangement of leaves on a stem, the spiral shells of snails, the segments of fruits like bananas and pineapples, and the proportions of the human body. The sequence is one of mathematics' most elegant bridges to the natural world.

Fibonacci numbers grow exponentially. The 50th Fibonacci number is 12,586,269,025. The 100th is approximately 3.54 * 10^20. This calculator handles large numbers efficiently but is designed for practical values of n (typically up to a few hundred).