The Fibonacci sequence is one of the most famous recursive sequences in mathematics.
Each term is obtained by adding the two previous terms.
1. Fibonacci sequence definition
This calculator uses the indexing convention:
\[
\begin{aligned}
F_0&=0,\\
F_1&=1.
\end{aligned}
\]
For every index \(k\ge2\), the next term is:
\[
\begin{aligned}
F_k&=F_{k-1}+F_{k-2}.
\end{aligned}
\]
So the sequence begins:
\[
\begin{aligned}
0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ldots
\end{aligned}
\]
2. What “position” means
The position of a Fibonacci number is the index \(k\) where the value appears:
\[
\begin{aligned}
F_k&=n.
\end{aligned}
\]
For example:
\[
\begin{aligned}
F_{10}&=55.
\end{aligned}
\]
Therefore, \(55\) is the 10th Fibonacci number when indexing starts at \(F_0=0\).
3. The special case of \(1\)
The value \(1\) appears twice near the beginning:
\[
\begin{aligned}
F_1&=1,\\
F_2&=1.
\end{aligned}
\]
Therefore, this calculator reports both positions for \(1\): \(F_1\) and \(F_2\).
4. Checking whether a number is Fibonacci
To check whether a nonnegative integer \(n\) is Fibonacci, generate terms from the beginning:
\[
\begin{aligned}
F_0,\ F_1,\ F_2,\ F_3,\ldots
\end{aligned}
\]
Stop when either:
- a generated term equals \(n\), meaning \(n\) is a Fibonacci number;
- a generated term becomes greater than \(n\), meaning \(n\) was skipped and is not a Fibonacci number.
5. Worked example: \(55\)
Generate the Fibonacci terms:
\[
\begin{aligned}
F_0&=0,\\
F_1&=1,\\
F_2&=1,\\
F_3&=2,\\
F_4&=3,\\
F_5&=5,\\
F_6&=8,\\
F_7&=13,\\
F_8&=21,\\
F_9&=34,\\
F_{10}&=55.
\end{aligned}
\]
Since \(F_{10}=55\), the number \(55\) is a Fibonacci number.
\[
\begin{aligned}
\boxed{55=F_{10}}.
\end{aligned}
\]
6. Worked example: \(54\)
Around \(54\), the Fibonacci sequence has:
\[
\begin{aligned}
F_9&=34,\\
F_{10}&=55.
\end{aligned}
\]
Since
\[
\begin{aligned}
34<54<55,
\end{aligned}
\]
no Fibonacci term equals \(54\). Therefore, \(54\) is not a Fibonacci number.
7. Generating Fibonacci numbers up to a limit
The calculator can also generate all Fibonacci terms up to a limit \(L\).
It keeps adding terms while:
\[
\begin{aligned}
F_k\le L.
\end{aligned}
\]
The list stops when the next Fibonacci term is greater than the limit.
8. Example: generate up to \(100\)
The Fibonacci terms not exceeding \(100\) are:
\[
\begin{aligned}
0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89.
\end{aligned}
\]
The next term is:
\[
\begin{aligned}
144>100.
\end{aligned}
\]
Therefore, the generation stops at \(89\).
9. Why the iterative method works
Fibonacci terms are nondecreasing after the start of the sequence.
Once the generated terms pass the target number \(n\), the target cannot appear later.
For example, if the sequence has already reached \(55\), then all later terms are at least \(55\), then larger after that.
So if a target such as \(54\) was not found before \(55\), it will never be found later.
10. Common examples
The table below uses plain text formulas to avoid raw LaTeX inside table cells.
| Number |
Position result |
Classification |
| 0 |
F0 = 0 |
Fibonacci |
| 1 |
F1 = 1 and F2 = 1 |
Fibonacci |
| 8 |
F6 = 8 |
Fibonacci |
| 55 |
F10 = 55 |
Fibonacci |
| 54 |
F9 = 34 and F10 = 55, so 54 is skipped |
Not Fibonacci |
| 100 |
F11 = 89 and F12 = 144, so 100 is skipped |
Not Fibonacci |
11. Formula summary
This table uses plain text formulas in table cells for reliable rendering across templates.
12. Common mistakes
- Forgetting that indexing can start at \(F_0=0\), not \(F_1=1\).
- Forgetting that \(1\) appears twice: \(F_1=1\) and \(F_2=1\).
- Calling a number Fibonacci just because it is close to a Fibonacci number.
- Stopping before the generated sequence passes the target.
- Mixing the value of a Fibonacci term with its index.
Key idea: generate Fibonacci terms until the target is found or passed. A match gives the position; passing the target means it is not Fibonacci.
