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.