This Fibonacci calculator is a tool for calculating the arbitrary terms of the Fibonacci sequence. Never again will you have to add the terms manually - our calculator finds the first 200 terms for you! You can also set your own starting values of the sequence and let this calculator do all work for you. Make sure to check out the geometric sequence calculator, too!

## What is the Fibonacci sequence?

The Fibonacci sequence is a sequence of numbers that follow a certain rule: each term of the sequence is equal to the sum of two preceding terms. This way, each term can be expressed by this equation:

`xₐ = xₐ₋₁ + xₐ₋₂`

The Fibonacci sequence typically has first two terms equal to x₀ = 0 and x₁ = 1. Alternatively, you can choose x₁ = 1 and x₂ = 1 as the sequence starters. Unlike in an arithmetic sequence, you need to know at least two consecutive terms to figure out the rest of the sequence.

The Fibonacci sequence rule is also valid for negative terms - for example, you can find x₋₁ to be equal to 1.

The first fifteen terms of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

## Formula for n-th term

Fortunately, calculating the n-th term of a sequence does not require you to calculate all of the preceding terms. There exists a simple formula that allows you to find an arbitrary term of the sequence:

`xₐ = (φⁿ - ψⁿ) / √5`

where:

- xₐ is the n-th term of the sequence,
- φ is the golden ratio (equal to (1 + √5)/2, or 1.618...)
`ψ = 1 - φ = (1 - √5)/2`

.

Our Fibonacci calculator uses this formula to find arbitrary terms in a blink of an eye!

## Formula for n-th term with arbitrary starters

You can also use the Fibonacci sequence calculator to find an arbitrary term of a sequence with different starters. Simply open the advanced mode and set two numbers for the first and second term of the sequence.

The Fibonacci calculator uses the following generalized formula for determining the n-th term:

`xₐ = aφⁿ + bψⁿ`

where:

`a = (x₁ - x₀ψ) / √5`

`b = (φx₀ - x₁) / √5`

- x₀ is the first term of the sequence,
- x₁ is the second term of the sequence.

## Negative terms of the Fibonacci sequence

If you write down a few negative terms of the Fibonacci sequence, you will notice that the sequence below zero has almost the same numbers as the sequence above zero. You can use the following equation to quickly calculate the negative terms:

`x₋ₐ = xₐ * (-1)ⁿ⁺¹`

For example, `x₋₈ = x₈ * (-1)⁸⁺¹ = x₈ * (-1) = -21`

## Fibonacci spiral

If you draw squares with sides of length equal to each consecutive term of the Fibonacci sequence, you can form a Fibonacci spiral:

The spiral in the image above uses the first ten terms of the sequence - 0 (invisible), 1, 1, 2, 3, 5, 8, 13, 21, 34.