The Fibonacci sequence
Each Fibonacci number is the sum of the two before it, starting from 1 and 1: 1, 1, 2, 3, 5, 8, 13, and onward. This calculator gives the nth number, the running total of the first n, the ratio between consecutive terms, and a preview of the sequence — with exact big-number arithmetic so large terms stay precise.
How it builds
Starting from two 1s, it repeatedly adds the last pair to get the next term. The sum of the first n terms has a neat shortcut — it equals the (n plus 2)th Fibonacci number minus 1 — so the totals grow just as fast as the sequence itself.
The golden ratio connection
Divide any term by the one before it and the result drifts toward 1.6180339, the golden ratio. The further out you go, the closer it gets, which is why Fibonacci numbers turn up in spirals, phyllotaxis, and proportions that feel naturally balanced.
Related tools
For sequences that add a fixed step, use the arithmetic sequence calculator; for ones that multiply by a fixed ratio, the geometric sequence calculator.
Worked example
The first ten terms are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, so F(10) is 55. They sum to 143, and 55 divided by 34 is about 1.6176 — already close to the golden ratio.
FAQ
Does it start at 0 or 1?
This calculator uses the common 1, 1 start, so F(1) and F(2) are both 1. Some texts begin with a 0; shift the index by one to match that convention.
How are very large terms kept exact?
It uses big-integer arithmetic, so even the 1000th term — hundreds of digits long — is computed with no rounding.