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 arithmetic 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:
If we write the terms of the series in the reverse order, the sum of terms remains the same; that is
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
Sum up the following series:
Here , ,
To find first, we have
Now using, we have
Find the sum of the first terms of the arithmetic series . Also find the sum of the first terms.
If is the sum of the first terms, then
For the sum of the first terms, we put in (1), i.e.