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.
