Definition of geometric progression

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.

If we consider the succession with first terms: $$a=(3,6,12,24,48,\ldots)$$ and we calculate the quotient of every term by the previous one, $$\dfrac{a_2}{a_1}=\dfrac{6}{3}=2,$$ $$\dfrac{a_3}{a_2}=\dfrac{12}{6}=2,$$ $$\dfrac{a_4}{a_3}=\dfrac{24}{12}=2,$$ $$\dfrac{a_5}{a_4}=\dfrac{48}{24}=2.$$

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 $(a_n)_{n\in\mathbb{N}}$, is a succession in which the quotient between two consecutive terms is constant, that is to say:

$$\dfrac{a_{n+1}}{a_n}=r$$

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

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

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

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