In mathematics, an

If the initial term of an arithmetic progression is and the common difference of successive members is

The behavior of the arithmetic progression depends on the common difference

The sum of the members of a finite arithmetic progression is called an

This is a generalization from the fact that the product of the progression is given by the factorial and that the product

**arithmetic progression**(AP) or**arithmetic sequence**is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2.If the initial term of an arithmetic progression is and the common difference of successive members is

*d*, then the*n*th term of the sequence () is given by:**finite arithmetic progression**and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an**arithmetic series**.The behavior of the arithmetic progression depends on the common difference

*d*. If the common difference is:- Positive, the members (terms) will grow towards positive infinity.
- Negative, the members (terms) will grow towards negative infinity.

## Sum

This section is about Finite arithmetic series. For Infinite arithmetic series, see Infinite arithmetic series.

**arithmetic series**. For example, consider the sum:*n*of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:### Derivation

To derive the above formula, begin by expressing the arithmetic series in two different ways:*d*cancel:*Aryabhatiya*(section 2.18).^{}## Product

The product of the members of a finite arithmetic progression with an initial element*a*_{1}, common differences*d*, and*n*elements in total is determined in a closed expressionThis is a generalization from the fact that the product of the progression is given by the factorial and that the product

*a*_{n}= 3 + (*n*-1)(5) up to the 50th term is## Standard deviation

The standard deviation of any arithmetic progression can be calculated via:**For detail Please Join Ajit Mishra's Online Classroom by****CLICK HERE**