In mathematics, an 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 nth term of the sequence () is given by:
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:
This is a generalization from the fact that the product of the progression is given by the factorial and that the product
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If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence () is given by:
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.
Derivation
To derive the above formula, begin by expressing the arithmetic series in two different ways:Product
The product of the members of a finite arithmetic progression with an initial element a1, 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
Standard deviation
The standard deviation of any arithmetic progression can be calculated via:For detail Please Join Ajit Mishra's Online Classroom by
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