Fibonacci Sequence Calculator

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About the Fibonacci Sequence Calculator

The fibonacci sequence calculator lets you calculate the nth\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 n\hspace{0.2em} n \hspace{0.2em} must be positive integers less than or equal to 1000\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 209\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

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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 0\hspace{0.2em} 0 \hspace{0.2em} (zeroth term) and 1\hspace{0.2em} 1 \hspace{0.2em} (first term).

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

The first few terms of the Fibonacci sequence are —

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

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

F0=0F_0 = 0 \hspace{0.2em},   F1=1\hspace{0.2em} F_1 = 1 \hspace{0.2em},   and   Fn+1=Fn+Fn1(n>1)\hspace{0.2em} F_{n + 1} = F_n + F_{n - 1} \hspace{0.5em} (n > 1) \hspace{0.2em}

For example,

F2=F1+F0=1+0=1\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*}


F3=F2+F1=1+1=2\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 Fn,n<0\hspace{0.2em} F_n \, , \hspace{0.5em} n < 0 \hspace{0.2em}). As an example,

F1=F0+F11=0+F1F1=1\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, F84\hspace{0.2em} F_{84} \hspace{0.2em} — the 84th\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, Fn\hspace{0.2em} F_n \hspace{0.2em}.

Fn=φnψn5F_n \hspace{0.25em} = \hspace{0.25em} \frac{\varphi^n - \psi^n}{\sqrt{5}}

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

φ=1+521.61803\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 ψ\hspace{0.2em} \psi \hspace{0.2em} is its conjugate.

ψ=1520.61803\begin{align*} \psi \hspace{0.25em} &= \hspace{0.25em} \frac{1 - \sqrt{5}}{2} \\[1.5em] &\approx \hspace{0.25em} -0.61803 \end{align*}

First 100\hspace{0.2em} 100 \hspace{0.2em} Fibonacci Terms

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