Fibonacci Sequence Calculator

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\hspace{0.2em} \,(n) \hspace{0.2em}

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About the Fibonacci Sequence Calculator

The fibonacci sequence calculator lets you calculate the $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$ term in the fibonacci sequence. You can also use it to get the numbers in the fibonacci sequence between two positions. Or check whether a number appears in the sequence.

Usage Guide

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i. Valid Inputs

The positions on the fibonacci terms $\hspace{0.2em} n \hspace{0.2em}$ must be positive integers less than or equal to $\hspace{0.2em} 1000 \hspace{0.2em}$ .

In the third calculator, the number to check (whether it apperars in the fibonacci sequence) must be a positive integer with upto $\hspace{0.2em} 209 \hspace{0.2em}$ digits.

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what the fibonacci sequence is.

Fibonacci Sequence

Fibonacci sequence is a sequence in which each term (except the first two) is the sum of the two preceding terms. Most commonly, the first two terms are taken to be $\hspace{0.2em} 0 \hspace{0.2em}$ (zeroth term) and $\hspace{0.2em} 1 \hspace{0.2em}$ (first term).

The terms in a Fibonacci sequence are known as Fibonacci numbers and are denoted by $\hspace{0.2em} F_n \hspace{0.2em}$ .

The first few terms of the Fibonacci sequence are —

$\hspace{0.2em} 0 \hspace{0.2em}$ , $\hspace{0.2em} 1 \hspace{0.2em}$ , $\hspace{0.2em} 1 \hspace{0.2em}$ , $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} 3 \hspace{0.2em}$ , $\hspace{0.2em} 5 \hspace{0.2em}$ , $\hspace{0.2em} 8 \hspace{0.2em}$ , and $\hspace{0.2em} 13 \hspace{0.2em}$

Put differently, if $\hspace{0.2em} F_n \hspace{0.2em}$ is the $\hspace{0.2em} n^{\text{th}} \hspace{0.2em}$ term of the Fibonacci sequence —

$F_0 = 0 \hspace{0.2em}$ , $\hspace{0.2em} F_1 = 1 \hspace{0.2em}$ , and $\hspace{0.2em} F_{n + 1} = F_n + F_{n - 1} \hspace{0.5em} (n > 1) \hspace{0.2em}$

For example,

\begin{align*} F_2 \hspace{0.25em} &= \hspace{0.25em} F_1 + F_0 \\[1em] &= \hspace{0.25em} 1 + 0 \\[1em] &= \hspace{0.25em} 1 \end{align*}

Similarly,

\begin{align*} F_3 \hspace{0.25em} &= \hspace{0.25em} F_2 + F_1 \\[1em] &= \hspace{0.25em} 1 + 1 \\[1em] &= \hspace{0.25em} 2 \end{align*}

We can use this formula to extend the fibonacci sequence backwards too (meaning, to find $\hspace{0.2em} F_n \, , \hspace{0.5em} n < 0 \hspace{0.2em}$ ). As an example,

\begin{align*} F_1 \hspace{0.25em} &= \hspace{0.25em} F_0 + F_{-1} \\[1em] 1 \hspace{0.25em} &= \hspace{0.25em} 0 + F_{-1} \\[1em] F_{-1} \hspace{0.25em} &= \hspace{0.25em} 1 \end{align*}

Alternate Formula

The formula discussed above may not be the super helpful if we want the value of anything beyond the first few terms, say, $\hspace{0.2em} F_{84} \hspace{0.2em}$ — the $\hspace{0.2em} 84^{\text{th}} \hspace{0.2em}$ fibonacci term.

Thankfully, we have a formula that gives us the value for any arbitrary fibonacci term, $\hspace{0.2em} F_n \hspace{0.2em}$ .

F_n \hspace{0.25em} = \hspace{0.25em} \frac{\varphi^n - \psi^n}{\sqrt{5}}

Here $\hspace{0.2em} \varphi \hspace{0.2em}$ is the golden ratio.

\begin{align*} \varphi \hspace{0.25em} &= \hspace{0.25em} \frac{1 + \sqrt{5}}{2} \\[1.5em] &\approx \hspace{0.25em} 1.61803 \end{align*}

And $\hspace{0.2em} \psi \hspace{0.2em}$ is its conjugate.

\begin{align*} \psi \hspace{0.25em} &= \hspace{0.25em} \frac{1 - \sqrt{5}}{2} \\[1.5em] &\approx \hspace{0.25em} -0.61803 \end{align*}