Fibonacci Calculator
Compute the nth Fibonacci number and display the sequence up to that term.
About Fibonacci Calculator
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
Worked example
With the default values below, Fibonacci Calculator returns F(n) of 55. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Term Number (n)accepts 0–1,000
- 10
Results
- 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
- 1Recurrence
- 2F(10)= 55
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.