For understanding the term ‘Arithmetic Sequence’, first we have to understand what is the meaning if sequence.

**Sequence**

A sequence is a group of numbers, which are in order. For example; 3,5,7,9… and so on.

Each number in the sequence or group of numbers is called a term. Sometimes they are called “elements” or “members”. Now,

**What is Arithmetic Sequence?**

In this sequence, the difference between one term and the next follows a constant behaviour. In other words, in here we add the same value or term each time to the infinity.

**Example:**

1,4,7,13,16,19,20,25,… here, this sequence follows the difference of 3 between numbers. The pattern is continuous by adding three each time as shown below,

So, commonly we write a correct sequence like this, or the formula for the correct sequence is;

{a, a+d, a+2d, a+3d, …}

**In here,**

- ‘a’ represents the first term of the sequence, and
- ‘d’ represents the difference between the terms, which is called the (common difference) of the sequence.

**Example:** (Continued from above)

**1,4,7,13,16,19,20,25,…**

It has,

- ‘a’ = 1 (which is the 1st term)
- ‘d’ = 3 (which is the “common difference” between the terms)

We get,

Formula is : { a, a+d, a+2d, a+3d,…}

{ 1, 1+3, 1+2×3, 1+3×3,…}

{1,4,7,10,…}

**Rule**

We can also write ‘AS’ (Arithmetic Sequence) as a rule,

Xn = a + d(n-1)

We use “n-1” because in the first term the ‘d’ is not used

**Example** : Find the 9th term from this sequence.

3, 8, 13, 18, 23, 28, 33, 38, …

Now, this sequence here has a common difference of 5 between them.

The value of **d** and **a** are:

**d = 5**(the common difference between the terms)**a = 3**(the first term of the sequence)

Now, by using the formula,

Xn = a + d(n-1)

= 3 + 5(n-1)

= 3 + 5n – 5

= 5n – 2

hence, the 9th term is. Here, n = 9.

X9 = 5 x 9 – 2

= 43