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

In other words, quantities are said to be in Arithmetic Sequence, when they increase or decrease by a by 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 that which follows it. In the first of the above examples, 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 first term is , while the third term minus the second is , the difference so 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 , the third term is . In each of these terms, the coefficient of is less than the number of term. 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 , ,

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