# How do you determine whether the sequence 36, 27, 18, 9,... is geometric and if it is, what is the common ratio?

Feb 26, 2017

The series $36 , 27 , 18 , 9 , \ldots \ldots .$ is not a geometric sequence, it is in fact arithmetic sequence.

#### Explanation:

In a geometric sequence, ratio of a term and its immediately preceding term is always constant.

In other words, to determine whether a sequence ${a}_{1} , {a}_{2} , {a}_{3} , {a}_{4} , {a}_{5} , \ldots \ldots \ldots \ldots .$ is a geometric sequence or not, one should check the ratios ${a}_{2} / {a}_{1} , {a}_{3} / {a}_{2} , {a}_{4} / {a}_{3} , {a}_{5} / {a}_{4}$ and if they are all equal i.e.

${a}_{2} / {a}_{1} = {a}_{3} / {a}_{2} = {a}_{4} / {a}_{3} = {a}_{5} / {a}_{4}$, then the sequence ${a}_{1} , {a}_{2} , {a}_{3} , {a}_{4} , {a}_{5} , \ldots \ldots \ldots \ldots .$ is a geometric sequence.

Here in the series $36 , 27 , 18 , 9 , \ldots \ldots .$

the ratios are $\frac{27}{36} , \frac{18}{27} , \frac{9}{18}$, which can be simplified to $\frac{3}{4} , \frac{2}{3} , \frac{1}{2}$ and as ratios are different, the series $36 , 27 , 18 , 9 , \ldots \ldots .$ is not a Geometric sequence.

Here, in fact we have $27 - 36 = 18 - 27 = 9 - 18 = - 9$ and what we have is that they have common difference and hence the series $36 , 27 , 18 , 9 , \ldots \ldots .$ is an arithmetic sequence.