Fibonacci Calculator
Calculate any term in the Fibonacci sequence. Find the nth Fibonacci number or generate the sequence up to n terms.
Enter n to find the Fibonacci number.
Fibonacci Sequence Formula
F(0) = 0, F(1) = 1
F(n) = F(n−1) + F(n−2) for n ≥ 2
F(n) = F(n−1) + F(n−2) for n ≥ 2
Each number in the Fibonacci sequence is the sum of the two preceding numbers, starting from 0 and 1.
Binet's Formula (Closed Form)
F(n) = (φⁿ − ψⁿ) / √5
where φ = (1 + √5) / 2 ≈ 1.618 and ψ = (1 − √5) / 2 ≈ −0.618
where φ = (1 + √5) / 2 ≈ 1.618 and ψ = (1 − √5) / 2 ≈ −0.618
Example: The first 10 Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
Properties of the Fibonacci Sequence
- The ratio of consecutive terms approaches the golden ratio (φ ≈ 1.618).
- Every 3rd Fibonacci number is even.
- The sum of the first n Fibonacci numbers equals F(n+2) − 1.
- GCD(F(m), F(n)) = F(GCD(m, n)).
First 20 Fibonacci Numbers
| n | F(n) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
| 8 | 21 |
| 9 | 34 |
| 10 | 55 |
| 15 | 610 |
| 20 | 6765 |
FAQ
Where is the Fibonacci sequence found in nature?
Fibonacci numbers appear in flower petals, sunflower spirals, pinecone patterns, tree branching, and shell spirals. These patterns relate to efficient packing and growth.
What is the golden ratio?
The golden ratio (φ ≈ 1.6180339887) is the limit of the ratio F(n+1)/F(n) as n approaches infinity.
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Last updated: 2026-08-08