Quick answer
The Fibonacci sequence starts F(0) = 0, F(1) = 1 and each term is the sum of the two before it. F(10) = 55, and the first terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Enter a position n (up to 1000). The calculator returns F(n) and the opening terms of the sequence.
Related tools: Random Number Generator: Random integers in a range, optionally repeatable with a seed. Scientific Calculator: Evaluate expressions with functions, constants, powers and factorials. Roman Numeral Converter: Convert between numbers and Roman numerals (1–3999).
How to use the Fibonacci Sequence Calculator
- Enter position n (a whole number, between 0 and 1000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Position n | A whole number | ≥ 0, ≤ 1000 |
Formula and method
Formula
F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2)
IEEE-754 double precision; shown to at most 6 decimal places.
Iterative BigInt addition, exact for any n up to 1,000.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Position n = 10.
| Result | Value |
|---|---|
| F(10) | 55 |
| First terms | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 |
Example 2
Inputs: Position n = 50.
| Result | Value |
|---|---|
| F(50) | 12,586,269,025 |
| First terms | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 |
Example 3
Inputs: Position n = 0.
| Result | Value |
|---|---|
| F(0) | 0 |
| First terms | 0 |
The first line is the headline result (f(10)). Below it the fibonacci Sequence Calculator lists first terms, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Growth and the golden ratio
Each term is about 1.618 times the previous one, the golden ratio. Binet's closed form F(n) = (φⁿ − ψⁿ) ÷ √5 gives the same values, but exact integers are computed by addition to avoid floating-point rounding.
Large terms
F(100) is 354,224,848,179,261,915,075 (a 21-digit number), and F(1000) has 209 digits. The calculator returns exact integers, not rounded approximations.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Position n: entering "1001" shows "Position n must be at most 1000." beside that field and no result.
Choosing between the Fibonacci Sequence Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
| [Scientific Calculator](/math/scientific-calculator/) | expression and angle mode | Result |
| [Roman Numeral Converter](/math/roman-numeral-converter/) | convert and value | Roman numeral |
| [Prime Number Calculator](/math/prime-number-calculator/) | number | Result |
| [Factorial Calculator](/math/factorial-calculator/) | n | n! |
Assumptions and limitations
- F(0) = 0
- Up to n = 1000
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for position n to confirm this.
Frequently asked questions
Does it start at 0 or 1?
Here F(0) = 0, F(1) = 1, F(2) = 1.
What is F(20)?
F(20) is 6,765 with the convention F(0) = 0 and F(1) = 1.
Where does the sequence appear?
In spirals, trees and algorithm analysis.
What does the Fibonacci Sequence Calculator calculate?
The nth Fibonacci number and the sequence leading to it. It returns F(10) and First terms.
What formula does the Fibonacci Sequence Calculator use?
F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2). Iterative BigInt addition, exact for any n up to 1,000.
What do I need to enter in the fibonacci Sequence Calculator?
You enter Position n.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. F(0) = 0.
What are the limits of this calculator?
F(0) = 0. Up to n = 1000.
What is an example of a fibonacci Sequence result?
With position n 10, the f(10) is 55.
Related calculators
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Scientific Calculator — Evaluate expressions with functions, constants, powers and factorials.
- Roman Numeral Converter — Convert between numbers and Roman numerals (1–3999).
- Prime Number Calculator — Check whether a number is prime and find the next prime.
- Factorial Calculator — n! computed exactly.
- Complex Number Calculator — Add, subtract, multiply and divide complex numbers.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.