Geometric progression

A sequence of numbers each one of which is equal to the preceding one multiplied by a number $q\ne0$ (the denominator of the progression). A geometric progression is called increasing if $q>1$, and decreasing if $0<q<1$; if $q<0$, one has a sign-alternating progression. Any term of a geometric progression $a_j$ can be expressed by its first term $a_0$ and the denominator $q$ by the formula \begin{equation} a_j=a_0q^{j}. \end{equation} The sum of the first $n$ terms of a geometric progression (with $q\ne1$) is given by the formula \begin{equation} a_0+a_0q+a_0q^2+\dots+a_0q^{n-1}= S_n = a_0\frac{1-q^n}{1-q}= \frac{a_n-a_0}{q-1} \end{equation} If $|q|<1$, the sum $S_n$ tends to the limit $S=a_0/(1-q)$ as $n$ tends to infinity. This number $S$ is known as the sum of the infinitely-decreasing geometric progression.

The expression \begin{equation} a_0+a_0q+a_0q^2+\dots+a_0q^{n}+\dots, \end{equation} if $|q|<1$ is the simplest example of a convergent series — a geometric series; the number $a_0/(1-q)$ is the sum of the geometric series.

The term "geometric progression" is connected with the following property of any term of a geometric progression with positive terms: $a_n = \sqrt{a_{n-1}a_{n+1}}$, i.e. any term is the geometric mean of the term which precedes it and the term which follows it.