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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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