Fibonacci Calculator
Find the nth Fibonacci number and generate the first N terms.
See the golden ratio approximation (1.61803) and Fibonacci in nature, art, and spiral patterns.
The Fibonacci sequence is a series of numbers where each term equals the sum of the two preceding terms. It appears throughout nature, art, finance, and computer science — and is intimately connected to the golden ratio.
Recursive formula: F(n) = F(n−1) + F(n−2) with seed values F(0) = 0 and F(1) = 1
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …
Closed-form formula (Binet’s Formula): F(n) = (φⁿ − ψⁿ) ÷ √5 where φ (phi) = (1 + √5) ÷ 2 ≈ 1.61803398… (the golden ratio) and ψ (psi) = (1 − √5) ÷ 2 ≈ −0.61803398…
The Golden Ratio: As n increases, the ratio of consecutive Fibonacci numbers F(n+1) ÷ F(n) approaches φ ≈ 1.61803398874…
- 8 ÷ 5 = 1.600
- 13 ÷ 8 = 1.625
- 89 ÷ 55 = 1.6182…
- 144 ÷ 89 = 1.6180…
Worked example — find F(10): F(0)=0, F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8, F(7)=13, F(8)=21, F(9)=34, F(10) = 55
Binet check: φ¹⁰ = 122.99…, ψ¹⁰ = 0.0081… F(10) = (122.99 − 0.0081) ÷ √5 = 122.98 ÷ 2.2361 ≈ 55 ✓
Where Fibonacci appears in nature:
- Spiral arrangement of sunflower seeds
- Number of petals on most flowers (3, 5, 8, 13, 21)
- Spiral of nautilus shells
- Branching patterns in trees and bronchial tubes
- Pinecone and pineapple spiral counts
In finance: Fibonacci retracement levels (23.6%, 38.2%, 61.8%, 78.6%) are widely used in technical analysis as potential support and resistance zones — derived from ratios between Fibonacci numbers.