Fibonacci Calculator
Compute the nth Fibonacci number and display the sequence up to that term.
Inputs
Which Fibonacci number to compute (F(0)=0, F(1)=1, …). Max 1000.
F(n)
55
Digits
2
F(n)/F(n−1) → φ
≈ 1.6180339887
Sequence (first terms)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Step by step
Recurrence
F(n) = F(n−1) + F(n−2), F(0)=0, F(1)=1
F(10)
= 55
How it works
The Fibonacci sequence is defined by F(0) = 0, F(1) = 1, and F(n) = F(n−1) + F(n−2) for n ≥ 2. The ratio of consecutive terms converges to the golden ratio φ = (1+√5)/2 ≈ 1.618. This calculator uses BigInt for exact computation up to F(1000), which has 209 digits.
Formulas
Fibonacci recurrence
F(n) = F(n−1) + F(n−2), F(0) = 0, F(1) = 1
- n
- Term index (0-based)
Binet's formula (closed form)
F(n) = (φⁿ − ψⁿ) / √5, where φ = (1+√5)/2, ψ = (1−√5)/2
- φ
- Golden ratio ≈ 1.618
- ψ
- Conjugate ≈ −0.618
Frequently Asked Questions
What is the golden ratio?
The golden ratio φ = (1 + √5)/2 ≈ 1.6180339887. The ratio F(n)/F(n−1) approaches φ as n grows. It appears throughout mathematics, art and nature.
Why use BigInt instead of floating point?
Fibonacci numbers grow exponentially. F(79) already exceeds Number.MAX_SAFE_INTEGER (2⁵³−1), so standard JavaScript numbers lose precision. BigInt provides exact integer arithmetic regardless of size.