Fibonacci Calculator
Find the Nth Fibonacci number, generate sequences, check if a number is Fibonacci, and explore the golden ratio, Binet's formula, and Lucas numbers.
F(n) — Nth Fibonacci Number
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First N Fibonacci Numbers —
Extended More scenarios, charts & detailed breakdown ▾
F(n)
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F(n-1) —
F(n+1) —
Professional Full parameters & maximum detail ▾
Golden Ratio & Binet
Golden Ratio Approx F(n+1)/F(n) —
Binet's Formula Result —
Lucas & Even Fibonacci
First 10 Lucas Numbers —
Sum of Even Fibonacci (first N terms) —
How to Use This Calculator
- Enter a position n to find the Nth Fibonacci number instantly.
- Use the Sequence tab to generate the first N Fibonacci numbers.
- Use the Check tab to test if a number is a Fibonacci number.
- The Professional tab shows the golden ratio approximation, Binet's formula, Lucas numbers, and even Fibonacci sums.
Formula
Recurrence: F(n) = F(n−1) + F(n−2), F(0)=0, F(1)=1
Binet: F(n) = (φⁿ − ψⁿ) / √5 where φ = (1+√5)/2
Example
F(10) = 55. First 8: 0, 1, 1, 2, 3, 5, 8, 13. Golden ratio ≈ F(11)/F(10) = 89/55 ≈ 1.6182.
Frequently Asked Questions
- The Fibonacci sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... where each number is the sum of the two preceding ones: F(n) = F(n−1) + F(n−2), with F(0)=0, F(1)=1.
- Binet's formula gives F(n) = (φⁿ − ψⁿ) / √5, where φ = (1+√5)/2 ≈ 1.61803 (golden ratio) and ψ = (1−√5)/2 ≈ −0.61803. It computes any Fibonacci number directly without iteration.
- The golden ratio φ ≈ 1.61803398... is the limit of F(n+1)/F(n) as n → ∞. It appears throughout art, architecture, and nature, and satisfies φ² = φ + 1.
- A positive integer n is Fibonacci if and only if one or both of 5n²+4 or 5n²−4 is a perfect square. For example, n=8: 5×64+4=324=18², so 8 is Fibonacci.
- Lucas numbers follow the same recurrence as Fibonacci (L(n) = L(n−1) + L(n−2)) but start with L(0)=2, L(1)=1: 2, 1, 3, 4, 7, 11, 18, 29, 47, ...