# Fibonacci Calculator

URL: https://speedycalculator.com/calculators/math/fibonacci/
Last updated: 2026-10-25

> The Fibonacci calculator returns the nth Fibonacci number using Binet’s formula, along with its neighbours and the ratio between them, which approaches the golden ratio.

## Inputs

- Position (n)

## Results

- Fₙ
- Fₙ₋₁
- Fₙ₊₁
- Fₙ₊₁ ÷ Fₙ

## Formula

```
Fₙ = round(φⁿ ÷ √5),  φ = (1 + √5) ÷ 2
```

## Explanation

Each Fibonacci number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, …

### How to use the Fibonacci Calculator

- **Position (n)**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Binet’s formula gives Fₙ directly without listing all earlier terms.

### What each result means

- **Fₙ** = round([Golden ratio]^[Position (n)] ÷ sqrt(5))
- **Fₙ₋₁** = round([Golden ratio]^([Position (n)]-1) ÷ sqrt(5))
- **Fₙ₊₁** = round([Golden ratio]^([Position (n)]+1) ÷ sqrt(5))
- **Fₙ₊₁ ÷ Fₙ** = [Fₙ₊₁] ÷ [Fₙ]

## Worked example

Inputs — Position (n): 20.

Results — **Fₙ = 6765**; Fₙ₋₁ = 4181; Fₙ₊₁ = 10946; Fₙ₊₁ ÷ Fₙ = 1.6180339985.

## FAQ

**Why stop at n = 78?**
Larger Fibonacci numbers exceed the precision of standard floating-point numbers.

**Where does the golden ratio appear?**
The ratio of consecutive Fibonacci numbers approaches 1.618…

**What formula does the Fibonacci Calculator use?**
Fₙ = round(φⁿ ÷ √5),  φ = (1 + √5) ÷ 2.