Difference between revisions of "Arithmetic progression"
From Encyclopedia of Mathematics
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''arithmetic series of the first order'' | ''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 | + | 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 | in which the general term is | ||
| − | + | $$a_n=a+(n-1)d.$$ | |
A characteristic property of an arithmetic progression is | 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==== | ====Comments==== | ||
For results on prime numbers in arithmetic progressions see [[Distribution of prime numbers|Distribution of prime numbers]]. | For results on prime numbers in arithmetic progressions see [[Distribution of prime numbers|Distribution of prime numbers]]. | ||
Revision as of 08:39, 20 April 2012
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:
$$ S_n=\frac{(a_1+a_n)n}{2}.$$
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=24847
Arithmetic progression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_progression&oldid=24847