Fibonacci Calculator
Find the nth Fibonacci Number ⚡
Enter a position n to find the nth Fibonacci number and see the full sequence up to that point, along with the golden ratio approximation.
Enter a position n to find the nth Fibonacci number and see the full sequence up to that point, along with the golden ratio approximation.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting with 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
It was introduced to the Western world by the Italian mathematician Leonardo of Pisa — nicknamed Fibonacci — in his 1202 book Liber Abaci. The sequence had been known in Indian mathematics centuries earlier.
The Fibonacci sequence is defined by the recurrence relation:
F(n) = F(n−1) + F(n−2), where F(1) = 0 and F(2) = 1
Each term is simply the sum of the two terms before it. There is also a closed-form formula (Binet's formula) that can calculate F(n) directly without iteration, though it involves the golden ratio φ.
As the sequence progresses, the ratio of consecutive terms (F(n)/F(n−1)) converges on the golden ratio φ ≈ 1.6180339887. By F(20) the approximation is already accurate to 9 decimal places.
The golden ratio appears throughout nature — in the spiral of nautilus shells, the arrangement of sunflower seeds, the branching of trees, and the proportions of many plants. It is also found in classical architecture and art.
Does the Fibonacci sequence start at 0 or 1?
This calculator uses the convention F(1) = 0, F(2) = 1. Some sources start with F(1) = 1, F(2) = 1 — in that version the values are the same but the index is shifted by one. Both conventions are widely used.
What is the golden ratio?
The golden ratio φ (phi) ≈ 1.618033... is an irrational number with the property that a/b = (a+b)/a. It is the limit of the ratio of consecutive Fibonacci numbers as the sequence progresses to infinity.
Why is the limit set at 75?
Fibonacci numbers grow exponentially. F(75) already has 15 digits. Beyond around F(79), numbers exceed JavaScript's safe integer limit (2⁵³) and precision begins to be lost. For most purposes, the first 75 terms cover every practical use case.
Are all Fibonacci numbers prime?
No — most are composite. The Fibonacci primes (Fibonacci numbers that are also prime) include 2, 3, 5, 13, 89, 233, 1597... but there is no pattern to when they occur, and it is unknown whether there are infinitely many.
What is a Fibonacci spiral?
A Fibonacci spiral is constructed by drawing quarter-circles through squares whose side lengths are consecutive Fibonacci numbers. It approximates the golden spiral and appears frequently in nature — most famously in the nautilus shell.