Question

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

Answer 1

The series `36,27,18,9,.......` 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,.............` 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,.............` is a geometric sequence.

Here in the series `36,27,18,9,.......`

the ratios are `27/36,18/27,9/18`, which can be simplified to `3/4,2/3,1/2` and as ratios are different, the series `36,27,18,9,.......` 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,.......` is an arithmetic sequence.