Fibonacci Calculator

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, finding 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.

A note on precision. This calculator holds every term as an exact integer, so F(100) comes out as 354,224,848,179,261,915,075 with all 21 digits correct. That matters from F(79) upward, where the numbers pass the point at which an ordinary JavaScript number can hold them exactly. The true F(79) is 14,472,334,024,676,221 and a naive calculator returns …220, one short. The golden ratio figures alongside are still floating point, which is why the difference from phi bottoms out around 10⁻¹⁶ rather than continuing to shrink.