Fibonacci Calculator

Find exact Fibonacci numbers, inspect neighboring terms and explore the recurrence from F₀ = 0 and F₁ = 1.

Calculation inputs

Precision & interpretation

For large n the first 20 terms are plotted separately from the selected exact result. A compact heading never changes the full value available in the exact-result section and exports.

Calculation results

Fibonacci number Fₙ55F10 · exact integer

Previous term34

Next term Fₙ₊₁89

Digits2

Step-by-step solution

Choose the formula
$F₀ = 0; F₁ = 1; Fₙ = Fₙ₋₁ + Fₙ₋₂$
Substitute your inputs
$Find F10 using zero-based indexing.$
Calculate in order
$F10 = 34 + 21$
Write the result
$F10 = 55$

Recurrence explorer

First 11 terms from index 0. The selected F10 result is computed separately with exact integers.

F₀ = 0 is explicit. Long integer values are never computed with a rounded Binet formula.

Exact value & calculation details

Quantity	Value
F10	55
F9	34
F11	89

Understand the calculation

Concept & meaning

Indexing matters: this calculator uses F₀ = 0 and F₁ = 1. The next terms are obtained by adding the previous two, rather than multiplying by a fixed ratio.

Formula & special cases

Fast-doubling identities compute Fₙ exactly with arbitrary-size integers. The teaching diagram uses the ordinary recurrence for nearby nonnegative indices.

Precision & limitations

For large n the first 20 terms are plotted separately from the selected exact result. A compact heading never changes the full value available in the exact-result section and exports.

F_{n} = F_{n - 1} + F_{n - 2}

Worked example from your inputs

F10 · exact integer = 55

Frequently asked questions

Does the sequence start with 0 or 1?

This page uses the zero-index convention: F₀ = 0, F₁ = 1, F₂ = 1.

Are large Fibonacci numbers exact?

Yes. The selected result and neighboring terms are stored as arbitrary-size integers up to n = 5,000.

Why is the golden-ratio value approximate?

For increasing n, Fₙ₊₁/Fₙ approaches the golden ratio but is not exactly equal to it at a finite nonzero index.

What happens at n = 0?

The result is 0. The previous nonnegative-index term does not exist, and the consecutive ratio with F₀ as denominator is undefined.

Show all questions (5 more)Show fewer questions

What does the Fibonacci calculate?

Find exact Fibonacci numbers, inspect neighboring terms and explore the recurrence from F₀ = 0 and F₁ = 1.

Which values should I enter?

Use the labeled fields above. Index starts at 0; whole number up to 5,000. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

F₀ = 0; F₁ = 1; Fₙ = Fₙ₋₁ + Fₙ₋₂

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$OpenStax official logo$ OpenStax · Sequences and notationRecursive sequences and factorial notationRead the reference

References explain the mathematics; they do not imply institutional endorsement.