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Difference between revisions of "Arithmetic progression"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133401.png" />, which is known as the difference of this progression. Thus, each arithmetic progression has the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133402.png" /></td> </tr></table>
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$$a,a+d,a+2d,\ldots,$$
  
 
in which the general term is
 
in which the general term is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133403.png" /></td> </tr></table>
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$$a_n=a+(n-1)d.$$
  
 
A characteristic property of an arithmetic progression is
 
A characteristic property of an arithmetic progression is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133404.png" /></td> </tr></table>
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133405.png" />, the progression is increasing; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133406.png" />, it is decreasing. The simplest example of an arithmetic progression is the series of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133407.png" />. The number of terms of an arithmetic progression can be bounded or unbounded. If an arithmetic progression consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133408.png" /> terms, its sum can be calculated by the formula:
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$$a_n=\frac{a_{n+1}+a_{n-1}}{2}.$$
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013340/a0133409.png" /></td> </tr></table>
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$$ S_n=\frac{(a_1+a_n)n}{2}.$$
  
 
====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]].

Latest revision as of 19:19, 7 February 2013

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:

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

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=12826