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The sequence of values of a polynomial of degree $m$:
 
The sequence of values of a polynomial of degree $m$:
  
$$p(x)=a_0+a_1x+\ldots+a_mx^m,$$
+
$$p(x)=a_0+a_1x+\dotsb+a_mx^m,$$
  
assumed by the polynomial when the variable $x$ takes successive integral non-negative values $x=0,1,\ldots$. If $m=1$, i.e. $p(x)=a_0+a_1x$, one obtains an arithmetic progression with initial term $a_0$ and difference $a_1$. If $p(x)=x^2$ or $p(x)=x^3$, one obtains sequences of squares or cubes of integers, i.e. special cases of arithmetic series of the second and third orders. If a first difference series is created, constituted by the differences between successive terms of an arithmetic series, then a series of differences of the first difference series (second differences) is written, and then the second differences are used to form third differences, etc., one finds, at the $m$-th stage, that all $m$-th differences are equal. Conversely, if the $m$-th differences of a numerical sequence are all equal, the sequence is an arithmetic series of order $m$. Using this property, it is possible to construct arithmetic series of different orders from their differences. For example, the sequence $1,1,1,\ldots,$ may be regarded as the first differences of the series of natural numbers $1,2,3,\ldots$; as the second differences of the series of triangular numbers $1,3,6,10,\ldots$; as the third differences of the sequence of tetrahedral numbers $1,4,10,20,\ldots,$ etc. These numbers are so called because triangular numbers represent numbers arranged in the form of a triangle (Fig. a), while tetrahedral numbers represent numbers arranged in the form of tetrahedra (pyramids) (Fig. b).
+
assumed by the polynomial when the variable $x$ takes successive integral non-negative values $x=0,1,\dotsc$. If $m=1$, i.e. $p(x)=a_0+a_1x$, one obtains an arithmetic progression with initial term $a_0$ and difference $a_1$. If $p(x)=x^2$ or $p(x)=x^3$, one obtains sequences of squares or cubes of integers, i.e. special cases of arithmetic series of the second and third orders. If a first difference series is created, constituted by the differences between successive terms of an arithmetic series, then a series of differences of the first difference series (second differences) is written, and then the second differences are used to form third differences, etc., one finds, at the $m$-th stage, that all $m$-th differences are equal. Conversely, if the $m$-th differences of a numerical sequence are all equal, the sequence is an arithmetic series of order $m$. Using this property, it is possible to construct arithmetic series of different orders from their differences. For example, the sequence $1,1,1,\dotsc,$ may be regarded as the first differences of the series of natural numbers $1,2,3,\dotsc$; as the second differences of the series of triangular numbers $1,3,6,10,\dotsc$; as the third differences of the sequence of tetrahedral numbers $1,4,10,20,\dotsc,$ etc. These numbers are so called because triangular numbers represent numbers arranged in the form of a triangle (Fig. a), while tetrahedral numbers represent numbers arranged in the form of tetrahedra (pyramids) (Fig. b).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013370a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013370a.gif" />
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Triangular numbers are expressed by the formula $[n(n+1)]/2$, while tetrahedral numbers are given by the formula
 
Triangular numbers are expressed by the formula $[n(n+1)]/2$, while tetrahedral numbers are given by the formula
  
$$\frac{n(n+1)(n+2)}{6},\quad n=1,2,\ldots.$$
+
$$\frac{n(n+1)(n+2)}{6},\quad n=1,2,\dotsc.$$
  
 
A generalization of triangular numbers is constituted by $k$-gonal or figurate numbers, which played an important role in the development of arithmetic in its various stages.
 
A generalization of triangular numbers is constituted by $k$-gonal or figurate numbers, which played an important role in the development of arithmetic in its various stages.
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$k$-gonal numbers are given by the formula:
 
$k$-gonal numbers are given by the formula:
  
$$n+(k-2)\frac{n(n-1)}{2},\quad n=1,2,\ldots.$$
+
$$n+(k-2)\frac{n(n-1)}{2},\quad n=1,2,\dotsc.$$
  
They form an arithmetic series of the second order, with one as their first term, $k$ as their second term and $k-2$ as their second differences. If $k=3$, triangular numbers are obtained; if $k=4$, one obtains squares ($n^2$); if $k=5$, one obtains pentagonal numbers $(3n^2-n)/2$, etc. These appellations will become clear from Fig. cand Fig. d, in which the number of beads arranged in the form of a square or a pentagon is expressed by the respective square or pentagonal number. Figurate numbers satisfy the following theorem, proposed by P. Fermat and first proved by A.L. Cauchy: Any natural number can be represented as a sum of not more than $k$ $k$-gonal numbers.
+
They form an arithmetic series of the second order, with one as their first term, $k$ as their second term and $k-2$ as their second differences. If $k=3$, triangular numbers are obtained; if $k=4$, one obtains squares ($n^2$); if $k=5$, one obtains pentagonal numbers $(3n^2-n)/2$, etc. These appellations will become clear from Fig. c and Fig. d, in which the number of beads arranged in the form of a square or a pentagon is expressed by the respective square or pentagonal number. Figurate numbers satisfy the following theorem, proposed by P. Fermat and first proved by A.L. Cauchy: Any natural number can be represented as a sum of not more than $k$ $k$-gonal numbers.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Arnol'd,  "Theoretical arithmetics" , Moscow  (1939)  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Arnol'd,  "Theoretical arithmetics" , Moscow  (1939)  (In Russian)</TD></TR>
 +
</table>
 +
 
 +
{{OldImage}}

Latest revision as of 07:28, 26 March 2023

of order $m$

The sequence of values of a polynomial of degree $m$:

$$p(x)=a_0+a_1x+\dotsb+a_mx^m,$$

assumed by the polynomial when the variable $x$ takes successive integral non-negative values $x=0,1,\dotsc$. If $m=1$, i.e. $p(x)=a_0+a_1x$, one obtains an arithmetic progression with initial term $a_0$ and difference $a_1$. If $p(x)=x^2$ or $p(x)=x^3$, one obtains sequences of squares or cubes of integers, i.e. special cases of arithmetic series of the second and third orders. If a first difference series is created, constituted by the differences between successive terms of an arithmetic series, then a series of differences of the first difference series (second differences) is written, and then the second differences are used to form third differences, etc., one finds, at the $m$-th stage, that all $m$-th differences are equal. Conversely, if the $m$-th differences of a numerical sequence are all equal, the sequence is an arithmetic series of order $m$. Using this property, it is possible to construct arithmetic series of different orders from their differences. For example, the sequence $1,1,1,\dotsc,$ may be regarded as the first differences of the series of natural numbers $1,2,3,\dotsc$; as the second differences of the series of triangular numbers $1,3,6,10,\dotsc$; as the third differences of the sequence of tetrahedral numbers $1,4,10,20,\dotsc,$ etc. These numbers are so called because triangular numbers represent numbers arranged in the form of a triangle (Fig. a), while tetrahedral numbers represent numbers arranged in the form of tetrahedra (pyramids) (Fig. b).

Figure: a013370a

Figure: a013370b

Triangular numbers are expressed by the formula $[n(n+1)]/2$, while tetrahedral numbers are given by the formula

$$\frac{n(n+1)(n+2)}{6},\quad n=1,2,\dotsc.$$

A generalization of triangular numbers is constituted by $k$-gonal or figurate numbers, which played an important role in the development of arithmetic in its various stages.

Figure: a013370c

Figure: a013370d

$k$-gonal numbers are given by the formula:

$$n+(k-2)\frac{n(n-1)}{2},\quad n=1,2,\dotsc.$$

They form an arithmetic series of the second order, with one as their first term, $k$ as their second term and $k-2$ as their second differences. If $k=3$, triangular numbers are obtained; if $k=4$, one obtains squares ($n^2$); if $k=5$, one obtains pentagonal numbers $(3n^2-n)/2$, etc. These appellations will become clear from Fig. c and Fig. d, in which the number of beads arranged in the form of a square or a pentagon is expressed by the respective square or pentagonal number. Figurate numbers satisfy the following theorem, proposed by P. Fermat and first proved by A.L. Cauchy: Any natural number can be represented as a sum of not more than $k$ $k$-gonal numbers.

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] V.I. Arnol'd, "Theoretical arithmetics" , Moscow (1939) (In Russian)


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How to Cite This Entry:
Arithmetic series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_series&oldid=32647
This article was adapted from an original article by BSE-2 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article