Fibonacci Calculator
Find the nth Fibonacci number, generate the first n terms, and see the golden ratio approximation.
Fibonacci Result
Fibonacci Sequence: Each number is the sum of the two preceding ones.
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)
First terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Golden Ratio (φ):
φ = (1 + √5) / 2 ≈ 1.6180339887...
As n grows, the ratio F(n)/F(n-1) converges to φ.
Binet’s Formula (closed-form):
F(n) = (φⁿ - ψⁿ) / √5
Where ψ = (1 - √5) / 2 ≈ -0.618
Fibonacci in nature:
- Sunflower seed spirals (21 and 34)
- Pinecone spirals (8 and 13)
- Nautilus shell chambers
- Tree branching patterns
Note: This calculator supports values up to n = 78 (before JavaScript loses integer precision).