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
Find the seventh term of an A.P in which the first term is and the common difference is .
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 .
Find the term of the following arithmetic progression:
Here , ,
Thus, the required term is .
Find the term of an A.P. whose term is and the term is .
Using , we have
Subtracting (1) and (2), we get
Putting the value in (1), we obtain
Putting , , in we get