# What is the nth Fibonacci number?

Finds the nth Fibonacci number F(n) exactly for n from 0 to 1,000, with the terms before it and the ratio of consecutive terms, which approaches the golden ratio.

- Page: https://www.acalculator.org/math/fibonacci-calculator
- JSON spec: https://www.acalculator.org/math/fibonacci-calculator.json
- Version: 34437026bda9

## Default answer

Example with the default inputs (Term number n 10): F(10) = 55.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n | Term number n | Which Fibonacci number to find, from F(0) = 0 to F(1000). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| value | F(n) | The nth Fibonacci number, exactly. |
| digits | Digits | How many digits F(n) has. |
| ratio | F(n) ÷ F(n − 1) | The ratio of the last two terms, rounded half up to 10 significant figures (from n = 2). |
| terms | Terms | The terms up to F(n), at most the last 20. |

## Method

F(0) = 0, F(1) = 1, F(n) = F(n − 1) + F(n − 2), added in exact whole numbers.

## Assumptions

- The sequence starts F(0) = 0, F(1) = 1, so F(1) = F(2) = 1 match OpenStax’s a₁ and a₂.
- n is a whole number from 0 to 1000.

## Worked examples

1. n = 10 gives value = 55, digits = 2, ratio = 1.617647059, terms = F(0) to F(10): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Source: OpenStax, Algebra and Trigonometry 2e, §13.1 Sequences and Their Notations (Fibonacci: a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂; a₁₀ = 34 + 21 = 55), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-1-sequences-and-their-notations.
2. n = 50 gives value = 12,586,269,025, digits = 11.
3. n = 100 gives value = 354,224,848,179,261,915,075, digits = 21, ratio = 1.618033989.
4. n = 0 gives value = 0, digits = 1, terms = F(0) to F(0): 0.

## FAQ

### What is the Fibonacci sequence?

Each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … The rule is F(n) = F(n − 1) + F(n − 2), starting from F(0) = 0 and F(1) = 1.

### What is the 10th Fibonacci number?

55. F(10) = F(9) + F(8) = 34 + 21 = 55. OpenStax starts the count at a₁ = 1, a₂ = 1 and also gets a₁₀ = 55, because F(1) = F(2) = 1 here too.

### Does the sequence start at 0 or 1?

Both are in use. This calculator uses F(0) = 0 and F(1) = 1, the common convention. From F(1) on, the terms match a sequence that starts 1, 1, so F(n) is the same in both counts for n ≥ 1.

### What does the ratio of Fibonacci numbers approach?

F(n) ÷ F(n − 1) approaches the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618033989. F(10) ÷ F(9) = 55 ÷ 34 ≈ 1.617647059, and by F(100) the ratio agrees with φ to 10 significant figures.

### Is there a formula for F(n) without adding up the terms?

Yes, Binet’s formula: F(n) = (φⁿ − ψⁿ) ÷ √5 with ψ = (1 − √5) ÷ 2. With floating point it loses digits for large n, so this calculator adds the terms in exact whole numbers instead.

### How large can n be?

n can be 0 to 1,000. F(1000) has 209 digits, and every digit is shown.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §13.1 Sequences and Their Notations (the Fibonacci sequence: a₁ = 1, a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3; 1, 1, 2, 3, 5, 8, 13, 21, 34, …; a₁₀ = 34 + 21 = 55). https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-1-sequences-and-their-notations (retrieved 2026-10-05)