Geometric progression: definition

A geometric progression is a type of succession, i.e., a sorted and infinite collection of real numbers, in which every term is obtained by multiplying its previous term by a constant value.

Example

If we consider the succession with first terms: $a = (3, 6, 12, 24, 48, \dots)$ and we calculate the quotient of every term by the previous one, $\frac{a_{2}}{a_{1}} = \frac{6}{3} = 2,$ $\frac{a_{3}}{a_{2}} = \frac{12}{6} = 2,$ $\frac{a_{4}}{a_{3}} = \frac{24}{12} = 2,$ $\frac{a_{5}}{a_{4}} = \frac{48}{24} = 2.$

We can see that this quotient is always the same number: $2$ . So we can define this succession recursively by multiplying by $2$ to obtain the next term.

Doing a formal definition, we will say that a geometric progression $(a_{n})_{n \in N}$ , is a succession in which the quotient between two consecutive terms is constant, that is to say:

$\frac{a_{n + 1}}{a_{n}} = r$

for any natural $n$ . We will call the constant $r$ ratio of the progression.

Example

The succession $(1, 3, 9, 27, 81, \dots)$ is a geometric succession of ratio $r = 3$ .

The succession $(\frac{1}{2}, 1, 2, 4, 8, \dots)$ is a geometric succession of ratio $r = 2$ .

The succession $(1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \dots)$ is a geometric succession of ratio $r = \frac{1}{4}$ .