# Arithmetic Sequence - Formula, Definition, Examples

TL;DR

An arithmetic sequence (or arithmetic progression, AP) is an ordered list of numbers where the difference between consecutive terms is constant — that constant is called the common difference (ddd). The nth-term formula is an=a1+(n−1)d and the sum of the first n terms is Sn=n2(a1+an).

## What Is an Arithmetic Sequence?

An **arithmetic sequence** is an ordered list of numbers where each term is obtained by adding a fixed constant to the previous one. That fixed constant is the **common difference**, denoted d.

Example sequence: 3,7,11,15,19,… — common difference d=4.

The general form is:

a1,;a1+d,;a1+2d,;a1+3d,;…

where a1 is the **first term** and d is the common difference. A sequence is arithmetic if and only if the difference between every pair of consecutive terms is the same.

## What Is the Arithmetic Sequence Formula?

There are two formulas you need.

### The nth-Term Formula

The n-th term is given by:

a_n = a_1 + (n - 1) d

where a1 is the first term, d is the common difference, and n is the position of the term you want.

**Example.** Find the 20th term of 5,8,11,14,… Here a1=5, d=3, n=20:

a_{20} = 5 + (20 - 1)(3) = 62

### The Sum Formula (Arithmetic Series)

The sum of the first n terms — called an **arithmetic series** and written Sn — has two equivalent forms:

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

**Example.** Sum the first 20 terms of 5,8,11,14,… We found a_{20} = 62:

S_{20} = \frac{20}{2}(5 + 62) = 670

## How Do You Find the Common Difference?

Subtract any term from the next one:

d = a_2 - a_1 = a_3 - a_2 = …

If the difference is the same every time, the sequence is arithmetic.

## What Is the Difference Between Arithmetic and Geometric Progressions?

An **arithmetic progression (AP)** adds a constant; a **geometric progression (GP)** multiplies by a constant.

| Feature                           | Arithmetic Progression (AP) | Geometric Progression (GP) |
|----------------------------------|------------------------------|------------------------------|
| Rule                             | Add a constant each step     | Multiply by a constant each step |
| Constant called                   | Common difference, d         | Common ratio, r             |
| nth term                        | a_n = a_1 + (n-1)d          | a_n = a_1 \cdot r^{n-1}   |
| Sum of first n                  | S_n = \frac{n}{2}(a_1 + a_n)| S_n = a_1 \cdot \frac{r^n - 1}{r - 1}, r≠1 |
| Growth shape                    | Linear                       | Exponential                 |

## Frequently Asked Questions

### What is an arithmetic sequence in simple words?
An arithmetic sequence is a list of numbers where each term is obtained by adding the same number to the previous one.

### What is the formula for an arithmetic sequence?
The n-th term: a_n = a_1 + (n−1)d, where a_1 is the first term, d is the common difference, and n is the position.

### How do you find the common difference?
Subtract any term from the next: d = a_2 - a_1.

### What is the difference between arithmetic and geometric sequence?
An arithmetic sequence adds the same number each step (common difference). A geometric sequence multiplies by the same number each step (common ratio).