A geometric sequence is a sequence in which the ratio $r$ between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with first term $3$ and common ratio $2.$ The behavior of geometric sequences generally depends on the values of the first term $a_1$ and the common ratio $r.$ The following table shows the effects of these parameters.

	$a_1>0$	$a_1<0$
$r>1$	Increasing $3\stackrel{\times 2}{\to }6\stackrel{\times 2}{\to }12\stackrel{\times 2}{\to }24\stackrel{\times 2}{\to }48\dots$	Decreasing $\text{-}3\stackrel{\times 2}{\to }\text{-}6\stackrel{\times 2}{\to }\text{-}12\stackrel{\times 2}{\to }\text{-}24\stackrel{\times 2}{\to }\text{-}48\dots$
$r=1$	Constant $3\stackrel{\times 1}{\to }3\stackrel{\times 1}{\to }3\stackrel{\times 1}{\to }3\stackrel{\times 1}{\to }3\dots$	Constant $\text{-}3\stackrel{\times 1}{\to }\text{-}3\stackrel{\times 1}{\to }\text{-}3\stackrel{\times 1}{\to }\text{-}3\stackrel{\times 1}{\to }\text{-}3\dots$
$0<r<1$	Decreasing $48\stackrel{\times \frac{1}{2}}{\to }24\stackrel{\times \frac{1}{2}}{\to }12\stackrel{\times \frac{1}{2}}{\to }6\stackrel{\times \frac{1}{2}}{\to }3\dots$	Increasing $\text{-}48\stackrel{\times \frac{1}{2}}{\to }\text{-}24\stackrel{\times \frac{1}{2}}{\to }\text{-}12\stackrel{\times \frac{1}{2}}{\to }\text{-}6\stackrel{\times \frac{1}{2}}{\to }\text{-}3\dots$
$r<0$	Alternating $3\stackrel{\times (\text{-}2)}{\to }\text{-}6\stackrel{\times (\text{-}2)}{\to }12\stackrel{\times (\text{-}2)}{\to }\text{-}24\stackrel{\times (\text{-}2)}{\to }48\dots$	Alternating $\text{-}3\stackrel{\times (\text{-}2)}{\to }6\stackrel{\times (\text{-}2)}{\to }\text{-}12\stackrel{\times (\text{-}2)}{\to }24\stackrel{\times (\text{-}2)}{\to }\text{-}48\dots$

Like for any other sequence, the first term of a geometric sequence is denoted by $a_1,$ the second by $a_2,$ and so on. Since geometric sequences have a common ratio $r,$ once one term is known, the next term can always be found by multiplying the known term by $r.$

In fact, the sequence can be found using only $a_1$ and $r,$ since all the subsequent terms can be found by multiplying $a_1$ by $r$ a specific number of times. Because of this, geometric sequences have the following general form.