Geometric progressions

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

$a(n)=c\,m^{n},\quad n=\{0,\ldots ,N-1\},\,$

where $\scriptstyle m\,$ and $\scriptstyle c\,$ are constants; therefore $\scriptstyle {\frac {a(n)}{a(n-1)}}\,=\,m\,$ and $\scriptstyle a(n)\,\equiv \,0{\pmod {m}},\,n\,\geq \,1.\,$ For example, {11, 44, 176, 704, 2816, 11264, 45056, ...} (A002089) is a geometric progression with $\scriptstyle m\,=\,4\,$ and $\scriptstyle c\,=\,11\,$. In terms of growth of sequences, nonconstant arithmetic progressions have exponential growth.

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

$a(n)={\sqrt[{2}]{a(n-1)\cdot a(n+1)}}=\exp {\left({\frac {\log a(n-1)+\log a(n+1)}{2}}\right)}.\,$

Recurrence

$a(n+1)=m\cdot a(n)\,$

Generating functions

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

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

See also

• Arithmetic progressions
• Harmonic progressions