# Arithmetic Sequence or Arithmetic Progression

An **arithmetic sequence or progression **(abbreviated as **A.P**) is a sequence in which each term after the first is obtained by adding a fixed number to the preceding term, which is called the **common difference**.

In other words, quantities are said to be in arithmetic sequence when they increase or decrease by a common difference. Thus each of the following series forms an arithmetic progression:

** **

The common difference is found by subtracting any term of the series from the term which follows it. In example 1 above, the common difference is ; in the second it is ; in the third it is .

But is not an A.P. Here the second term minus the first term is , while the third term minus the second is . The difference that is obtained does not remain the same.

__The nth term of an arithmetic progression__

Let be the first term and be the constant difference. Then the second term is , and the third term is . In each of these terms, the coefficient of is less than the number of terms. Similarly, the 10th term is . The nth term is the term after the first term and is obtained after has added times in succession. Hence, if represents the term, then

__Example__:

Find the seventh term of an A.P in which the first term is and the common difference is .

__Solution__:

The seventh term may be designed as , we use

as the formula and substitute for the variables to find.

Here , ,

Thus, the required seventh term is .

__Example__:

Find the term of the following arithmetic progression:

__Solution__:

Here , ,

This gives

Thus, the required term is .

** Example:
**Find the term of an A.P. whose term is and the term is .

__Solution__:

Using , we have

------- (1)

------- (2)

Subtracting (1) and (2), we get

Putting the value in (1), we obtain

Putting , , in we get