Fibonacci Number Calculator
Enter an index n to get the Fibonacci number Fn, plus the neighboring terms, the golden-ratio approximation, and the recent sequence.
Last updated:
Input
Enter the index n to compute the Fibonacci number Fn, along with the neighboring terms and the golden-ratio approximation.
Enter a non-negative integer (e.g. 10).
Result
Fibonacci number F(10)
55
Previous term F(10 - 1)
34
Next term F(10 + 1)
89
Golden-ratio approximation Fn / F(n-1)
1.617647058
As n grows, this ratio approaches the golden ratio phi ≈ 1.61803399.
Surrounding sequence
| Index | Value |
|---|---|
| F(3) | 2 |
| F(4) | 3 |
| F(5) | 5 |
| F(6) | 8 |
| F(7) | 13 |
| F(8) | 21 |
| F(9) | 34 |
| F(10) | 55 |
How it works
- The Fibonacci sequence sets F0=0 and F1=1, and every later term is the sum of the previous two: Fn=F(n-1)+F(n-2). It runs 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.
- To avoid overflow errors, this tool computes the terms with arbitrary-precision integers. Even for large n, Fn is shown as an exact integer.
- The ratio of consecutive terms Fn/F(n-1) approaches the golden ratio phi = (1+sqrt 5)/2 ≈ 1.6180339887 as n grows. The tool reports this approximation alongside the result.
- Fibonacci numbers appear in natural spirals such as leaf arrangements and sunflower seed patterns, and they are widely used in algorithms and art.
- The index n must be a non-negative integer. Negative values and decimals cannot be computed.
Frequently asked questions
- How is the Fibonacci number Fn defined?
- It starts with F0=0 and F1=1, and each later term is the sum of the two before it, Fn=F(n-1)+F(n-2). The sequence runs 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.
- How is Fibonacci related to the golden ratio?
- The ratio of consecutive terms Fn/F(n-1) approaches the golden ratio phi=(1+sqrt5)/2, about 1.6180339887, as n grows. The tool shows this approximation alongside the term.
- Does it stay exact for large n, and what inputs are allowed?
- Terms are computed step by step with big integers, so Fn is exact even for large n. The index n must be an integer of 0 or more; negatives and decimals are not accepted.
Reviews
Tell us what you think of this calculator.
Write a review
Found an incorrect result or a problem? Turn this on to report it.
Rating