# 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.

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 $$r=\dfrac{1}{4}$$.