Fibonacci Sequence Calculator
Generate the Fibonacci sequence, find any specific Fibonacci number up to F(75), and watch the ratio of consecutive terms converge to the golden ratio. A powerful tool for students and math enthusiasts.
The Fibonacci sequence is one of the most famous sequences in mathematics. Starting with 0 and 1, each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
The Golden Ratio
As the sequence progresses, the ratio of consecutive Fibonacci numbers approaches the golden ratio, phi = (1 + sqrt(5))/2 approximately equal to 1.6180339887. This irrational number appears throughout nature, art, and architecture.
Fibonacci in Nature
Fibonacci numbers appear in sunflower seed spirals, pinecone scales, tree branching patterns, and the shell of a nautilus. The golden ratio is found in the proportions of the Parthenon, the Mona Lisa, and modern design systems.
Mathematical Properties
Every third Fibonacci number is even. The sum of the first n Fibonacci numbers equals F(n+2) - 1. The GCD of two Fibonacci numbers F(m) and F(n) equals F(GCD(m,n)). Fibonacci numbers also connect to Pascal's triangle through diagonal sums.