Fibonacci Calculator
Find any Fibonacci number instantly. Enter a term number to get F(n), the running sum, and the full sequence with the classic recurrence explained.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Show the sequence up to F(n)
How the Fibonacci Calculator Works
The Fibonacci sequence is one of the most famous patterns in mathematics. Each number is the sum of the two before it. This calculator finds the nth term F(n), the sum of the first n+1 terms, and lists the sequence so you can see the pattern build.
The Formula
The sequence is defined by a simple recurrence relation:
F(0) = 0, F(1) = 1, and F(n) = F(n − 1) + F(n − 2) for n ≥ 2
In words: start with 0 and 1, then keep adding the last two numbers to get the next one. The tool iterates this rule using exact big-integer math so large terms stay precise.
Worked Example
To find F(10), build the sequence one term at a time:
| n | F(n) | How |
|---|---|---|
| 0 | 0 | start |
| 1 | 1 | start |
| 2 | 1 | 0 + 1 |
| 3 | 2 | 1 + 1 |
| 4 | 3 | 1 + 2 |
| 5 | 5 | 2 + 3 |
| 6 | 8 | 3 + 5 |
| 7 | 13 | 5 + 8 |
| 8 | 21 | 8 + 13 |
| 9 | 34 | 13 + 21 |
| 10 | 55 | 21 + 34 |
So F(10) = 55, and the sum of terms F(0) through F(10) is 143.
Where Fibonacci Numbers Appear
The sequence shows up far beyond textbooks:
- Nature — petal counts, pinecone spirals, and leaf arrangements often follow Fibonacci numbers.
- The golden ratio — consecutive terms approach roughly 1.618, a proportion used in art and design.
- Computer science — Fibonacci numbers appear in algorithm analysis, heaps, and search techniques.
Enter any term number to explore the sequence and confirm results instantly.
Frequently asked questions
What is the Fibonacci sequence?+
It is a series where each number is the sum of the two before it, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on.
What is the 10th Fibonacci number?+
Counting from F(0)=0, the 10th term F(10) is 55. The sequence up to it is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Does the Fibonacci sequence start at 0 or 1?+
Most mathematical definitions start at F(0)=0 and F(1)=1. This calculator uses that convention, so the term number you enter matches the standard index.
What is the golden ratio connection?+
As the sequence grows, the ratio between consecutive Fibonacci numbers gets closer and closer to the golden ratio, approximately 1.618.
Can this tool handle very large Fibonacci numbers?+
Yes. It uses exact big-integer arithmetic, so even large term numbers up to 1000 return the precise value without rounding errors.