# Fibonacci Calculator

Created by Bogna Szyk
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Jun 05, 2023

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 250 terms for you! You can also set your own starting values of the sequence and let this calculator do all the work for you.

If you want to learn about how stock traders apply Fibonacci in trading, check out the Fibonacci retracement calculator. Make sure to check out the geometric sequence calculator, too, or if you want some more "weird" sequences, the Collatz conjecture calculator!

## 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:

Fₙ = Fₙ₋₂ + Fₙ₋₁

The Fibonacci sequence typically has the first two terms equal to F₀ = 0 and F₁ = 1. Alternatively, you can choose F₁ = 1 and F₂ = 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 F₋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...

Interestingly, Fibonacci numbers follow the well-known Benford's law!

## 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:

Fₙ = (φⁿ - ψⁿ) / √5

where:

• Fₙ – n-th term of the sequence,
• φGolden 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 terms of the sequence.

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

Fₙ = aφⁿ + bψⁿ

where:

• a = (F₁ - F₀ψ) / √5
• b = (φF₀ - F₁) / √5
• F₀ – First term of the sequence,
• F₁ – Second term of the sequence, etc.

## 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. The difference is that the result is positive for odd negative values of n. You can use the following equation to quickly calculate the negative terms:

F₋ₙ = Fₙ × (-1)ⁿ⁺¹

For example, F₋₈ = F₈ × (-1)⁸⁺¹ = F₈ × (-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.

## FAQ

### How do you get Fibonacci numbers?

1. Pick 0 and 1. Then you sum them, and you have 1. Look at the series you built: 0, 1, 1.
2. For the 3rd number, sum the last two numbers in your series; that would be 1+1. Now your series looks like 0, 1, 1, 2.
3. For the 4th number of your Fibo series, sum the last two numbers: 2+1 (note you picked the last two numbers again). Your series: 0, 1, 1, 2, 3. And so on.

### What Fibonacci numbers used for?

Here are some of the most common applications for the Fibonacci numbers:

• For investing in the stock market. Investors believe stock prices move to respect certain Fibonacci levels.
• In music, specifically in western music, musicians use Fibonacci numbers for musical scales.

-Artists use it as an aesthetical concept based on the Fibonacci spiral.

### What are the first 10 Fibonacci numbers?

The first 10 Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. You can also check our Fibonacci calculator to get any number you want.

### How to calculate the golden ratio?

1. Pick a number. Let's say 2. Divide 1 by your value. In this case: 1/2 = 0.5
2. Add 1. Now you have 1.5. Repeat.
3. Divide 1 by 1.5: 1/1.5 or 0.6666. Add 1 again. You have 1.6666.
5. Repeat two more times, and you will get: 1.625 and 1.615. If you repeat it enough times, you will get 1.618.
Bogna Szyk
F0 = 0, F1 = 1,
Fn = Fn-2 + Fn-1
n
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