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Arithmetic progression

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2020 Mathematics Subject Classification: Primary: 11B25 [MSN][ZBL]

arithmetic series of the first order

A sequence of numbers in which each term is obtained from the term immediately preceding it by adding to the latter some fixed number , which is known as the difference of this progression. Thus, each arithmetic progression has the form

a,a+d,a+2d,\ldots,

in which the general term is

a_n=a+(n-1)d.

A characteristic property of an arithmetic progression is


a_n=\frac{a_{n+1}+a_{n-1}}{2}.

If d>0, the progression is increasing; if d<0, it is decreasing. The simplest example of an arithmetic progression is the series of natural numbers 1,2,\ldots. The number of terms of an arithmetic progression can be bounded or unbounded. If an arithmetic progression consists of n terms, its sum can be calculated by the formula:

S_n=\frac{(a_1+a_n)n}{2}.

Comments

For results on prime numbers in arithmetic progressions see Distribution of prime numbers.

How to Cite This Entry:
Arithmetic progression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_progression&oldid=29401