Fibonacci Calculator

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!

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:

F_n = F_n-2 + F_n-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!

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:

F_n = (φⁿ − ψⁿ) / √5

where:

• F_n — n-th term of the sequence;
• φ — Golden ratio (equal to (1 + √5)/2, or 1.618...); and
• ψ = 1 − φ = (1 − √5)/2.

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

You can also use the Fibonacci sequence calculator to find an arbitrary term of a sequence with different starters. Simply click on Modify F₀ and F₁ in the bottom part of the calculator, 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:

F_n = aφⁿ + bψⁿ

where:

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

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 = F_n × (-1)ⁿ⁺¹

For example, F₋₈ = F₈ × (-1)⁸⁺¹ = F₈ × (-1) = -21

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.