Number Sequence Calculator

Use this number sequence calculator to generate arithmetic, geometric, and Fibonacci sequences. Get step-by-step solutions and sequence sums instantly.

Number Sequence Calculator

Select the type of sequence, enter the required parameters, and click Calculate to generate the sequence and its sum.

Sequence Type

First Term (a₁)

Common Difference (d)

Number of Terms (n)

What is a Number Sequence?

A number sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. Sequences can be finite (with a limited number of terms) or infinite (continuing without end).

Common types of sequences include arithmetic sequences (where the difference between consecutive terms is constant), geometric sequences (where the ratio between consecutive terms is constant), and Fibonacci sequences (where each term is the sum of the two preceding terms).

Sequence Formulas

Here are the key formulas for different types of sequences:

Arithmetic Sequence

An arithmetic sequence has a constant difference between consecutive terms:

$a_n = a_1 + (n-1)d$ Where: - $a_n$ is the nth term - $a_1$ is the first term - $d$ is the common difference - $n$ is the position in the sequence

The sum of the first n terms of an arithmetic sequence is:

$S_n = \frac{n}{2}[2a_1 + (n-1)d]$ or $S_n = \frac{n}{2}(a_1 + a_n)$

Geometric Sequence

A geometric sequence has a constant ratio between consecutive terms:

$a_n = a_1 \cdot r^{n-1}$ Where: - $a_n$ is the nth term - $a_1$ is the first term - $r$ is the common ratio - $n$ is the position in the sequence

The sum of the first n terms of a geometric sequence is:

$S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$ $S_n = n \cdot a_1$ for $r = 1$

Fibonacci Sequence

In a Fibonacci sequence, each term is the sum of the two preceding terms:

$F_n = F_{n-1} + F_{n-2}$ With starting values typically $F_1 = 0$ and $F_2 = 1$

Applications of Number Sequences

Mathematics

Used in number theory, calculus, and mathematical modeling. Sequences form the foundation for series, which are essential in advanced mathematics.

Physics

Applied in wave theory, quantum mechanics, and describing natural phenomena like planetary orbits and oscillations.

Finance

Used in compound interest calculations, loan amortization, and financial forecasting models.

Computer Science

Essential in algorithm analysis, data structures, and recursive functions. The Fibonacci sequence is particularly important in computer programming.

How to Use This Calculator

Select the type of sequence you want to generate (arithmetic, geometric, or Fibonacci).
Enter the required parameters for your chosen sequence type.
Specify the number of terms you want to generate (maximum 100).
Click 'Calculate' to generate the sequence and see the step-by-step solution.