Arithmetic progression

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 $d$, 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:

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

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