Math

Fibonacci Number Calculator

Find the nth Fibonacci number, where each number is the sum of the two before it.

By TheCalcTools Updated Formula checked and tested

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F(n)
F(n)
—
F(n+1) ÷ F(n)
—

Fibonacci Number Calculator by TheCalcTools

How the Fibonacci Number Calculator works

Enter the position n (F(0) = 0, F(1) = 1, F(2) = 1, …).

The calculator uses Binet’s closed-form formula, rounded to the nearest whole number.

The ratio of consecutive terms approaches the golden ratio, about 1.618.

The formula

F(n) = F(n−1) + F(n−2)F(n) ≈ φⁿ ÷ √5, where φ = (1 + √5) ÷ 2

Worked example

Using the default values shown in the calculator:

Position (n)20
  1. The sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
  2. F(20) = 6,765.
  3. F(21) ÷ F(20) = 1.6180, close to the golden ratio.

Result:

F(n)6,765
F(n+1) ÷ F(n)1.6180

Frequently asked questions

Why is n limited to 70?

Binet’s formula uses floating-point numbers. Up to n = 70 the rounded result is exact; beyond that precision is lost.

What is the golden ratio?

It is about 1.6180339887. The ratio of consecutive Fibonacci numbers converges to it.

Sources and further reading

Factorial CalculatorCalculate the factorial n! of a whole number Geometric Sequence CalculatorEnter the first term, common ratio and n to get the nth term, the partial sum and the… GCD and LCM CalculatorEnter two whole numbers to find their greatest common divisor and least common multiple Prime Factorization CalculatorEnter a whole number to break it into prime factors Prime Number Calculator (Is It Prime? Next and Previous Prime)Enter a whole number to test whether it is prime Factors Calculator (All Divisors of a Number)Enter a whole number to list all of its factors

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