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# Arithmetic progressions

${\displaystyle \scriptstyle N\,}$-terms arithmetic progressions are sequences of the form (with ${\displaystyle \scriptstyle N\,\to \,\infty \,}$ for an infinity of terms)

${\displaystyle a(n)=mn+c,\quad n=\{0,\ldots ,N-1\},\,}$

where ${\displaystyle \scriptstyle m\,}$ and ${\displaystyle \scriptstyle c\,}$ are constants; therefore ${\displaystyle \scriptstyle a(n)-a(n-1)\,=\,m\,}$ and ${\displaystyle \scriptstyle a(n)\,\equiv \,c{\pmod {m}}\,}$. For example, {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} (A017569) is an arithmetic progression with ${\displaystyle \scriptstyle m\,=\,12\,}$ and ${\displaystyle \scriptstyle c\,=\,4\,}$. In terms of growth of sequences, nonconstant arithmetic progressions have linear growth.

Equivalently, a sequence is an arithmetic progression when each term is the arithmetic mean of the neighboring terms, i.e.

${\displaystyle a(n)={\frac {a(n-1)+a(n+1)}{2}}.\,}$

## "Primitive" versus "nonprimitive" arithmetic progressions

An arithmetic progression might be said to be "primitive" if ${\displaystyle \scriptstyle m\,}$ and ${\displaystyle \scriptstyle c\,}$ are coprime. An arithmetic progression with ${\displaystyle \scriptstyle \gcd(m,\,c)\,=\,k,\,k\,>\,1\,}$ (cf. gcd), which might thus be said to be "nonprimitive", is ${\displaystyle \scriptstyle k\,}$ times the corresponding "primitive" arithmetic progression. For example {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} = 4 × {1, 4, 7, 10, 13, 16, 19, 22, 25, 28, ...}.

## Recurrence

${\displaystyle a(n+1)=a(n)+m\,}$

## Generating functions

Arithmetic progressions have rational [ordinary] generating functions of the form

${\displaystyle G_{\{mn+c\}_{n=1}^{\infty }}(x)\equiv \sum _{n=1}^{\infty }(mn+c)\,x^{n}=-\,{\frac {c\,x^{2}-(m+c)\,x}{(x-1)^{2}}}.\,}$