Fibonacci Calculator — Formule
## Fibonacci Sequence
### Definition
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
### Binet's Formula
**F(n) = (phi^n - psi^n) / sqrt(5)**
where phi = (1 + sqrt(5))/2 (golden ratio) and psi = (1 - sqrt(5))/2.
For large n, psi^n approaches 0, so F(n) ≈ phi^n / sqrt(5).
### Golden Ratio
As n grows, F(n)/F(n-1) approaches the golden ratio phi ≈ 1.6180339887.
### Definition
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
### Binet's Formula
**F(n) = (phi^n - psi^n) / sqrt(5)**
where phi = (1 + sqrt(5))/2 (golden ratio) and psi = (1 - sqrt(5))/2.
For large n, psi^n approaches 0, so F(n) ≈ phi^n / sqrt(5).
### Golden Ratio
As n grows, F(n)/F(n-1) approaches the golden ratio phi ≈ 1.6180339887.
Exemple Résolu
Find the 10th Fibonacci number.
- F(10) = phi^10 / √5
- = 1.61803^10 / 2.23607
- ≈ 122.992 / 2.236
- ≈ 55
- Sequence up to F(10): 0,1,1,2,3,5,8,13,21,34,55