Number Sequence Calculator

Calculate arithmetic, geometric, and Fibonacci sequence terms quickly and accurately.

▼ Enter the values and click Calculate

Arithmetic Sequence Calculator

Calculate the nth term of an arithmetic sequence.

First Term

Common Difference

Term Number (n)

Result —

Geometric Sequence Calculator

Calculate the nth term of a geometric sequence.

First Term

Common Ratio

Term Number (n)

Result —

Fibonacci Sequence Calculator

Calculate a term in the Fibonacci sequence.

Term Number (n)

Result —

What Is a Number Sequence?

A number sequence is an ordered list of numbers that follows a particular pattern. Each number in a sequence is called a term, and the number of terms is called the length of the sequence.

There are many different types of number sequences. Three common types are arithmetic sequences, geometric sequences, and Fibonacci sequences.

Arithmetic Sequence
Each term is found by adding a constant common difference to the previous term.

Geometric Sequence
Each term is found by multiplying the previous term by a constant common ratio.

Fibonacci Sequence
Each term after the first two is the sum of the two preceding terms.

Arithmetic Sequences

An arithmetic sequence is a number sequence in which the difference between each successive term remains constant. This constant is called the common difference.

Formula:
a_n = a₁ + d × (n − 1)

Example:
Sequence: 4, 9, 14, 19, 24, ...
First term = 4
Common difference = 5
5th term = 4 + 5 × (5 − 1) = 24

The common difference can be positive or negative, depending on whether the sequence increases or decreases.

Geometric Sequences

A geometric sequence is a number sequence in which each successive term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Formula:
a_n = a × r^{n − 1}

Example:
Sequence: 3, 6, 12, 24, 48, ...
First term = 3
Common ratio = 2
5th term = 3 × 2⁴ = 48

The common ratio determines how the sequence grows or decreases from one term to the next.

Fibonacci Sequence

A Fibonacci sequence is a sequence in which every number following the first two is the sum of the two preceding numbers. In this calculator, the sequence starts with 0 and 1.

Formula:
a_n = a_n−1 + a_n−2
a₀ = 0, a₁ = 1

Example:
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5

Each new term is produced by adding the two previous terms together.

How to Use the Number Sequence Calculator

Arithmetic Sequence: Enter the first term, common difference, and term number, then click Calculate.

Geometric Sequence: Enter the first term, common ratio, and term number, then click Calculate.

Fibonacci Sequence: Enter the term number and click Calculate. This calculator uses a₀ = 0 and a₁ = 1.

Frequently Asked Questions

What is a number sequence?

A number sequence is an ordered list of numbers that follows a particular pattern. Each number in the sequence is called a term.

What is an arithmetic sequence?

An arithmetic sequence is a sequence in which the difference between consecutive terms remains constant. This difference is called the common difference.

What is a geometric sequence?

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant common ratio.

What is the Fibonacci sequence?

The Fibonacci sequence is a sequence in which each term after the first two is the sum of the two preceding terms. This calculator starts with 0 and 1.

Can arithmetic sequences have a negative difference?

Yes. A negative common difference produces a decreasing arithmetic sequence.

Can geometric sequences have a negative ratio?

Yes. A negative common ratio causes the signs of the terms to alternate.

Can I calculate a large Fibonacci term?

The calculator can calculate Fibonacci terms using the standard sequence starting with 0 and 1. Very large term numbers may exceed the range of standard JavaScript numbers.

Disclaimer

This calculator is provided for general mathematical and educational purposes. While we aim to provide accurate results, SmartMyTools does not guarantee that the results are suitable for professional, academic, financial, engineering, or other specialized applications.