# Arithmetic Series

The sum of an indicated number of terms in a sequence is called a **series**. The series obtained by adding the terms of an arithmetic progression is called an **a****rithmetic series**.

For example, the sum of the first seven terms of the sequence is the series,

The above series is also called the partial sum of the sequence .

If the number of terms in a series is finite, then the series is called a finite series, while a series consisting of an unlimited number of terms is known as an infinite series.

__The sum of the first n terms of an arithmetic series__

For any sequence , we have

If is an A.P., then can be written with usual notation as:

--- (1)

If we write the terms of the series in the reverse order, the sum of terms remains the same; that is

--- (2)

Adding (1) and (2), we get

Thus, --- (3)

If in (3) we replace by its value as , then we obtain another useful rule for as

--- (4)

__Example__:

Sum up the following series:

__Solution__:

Here , ,

To find first, we have

Now using, we have

__Example__:

Find the sum of the first terms of the arithmetic series . Also find the sum of the first terms.

__Solution__:

Here and

If is the sum of the first terms, then

------ (1)

For the sum of the first terms, we put in (1), i.e.