The key thing to look for to determine whether a sequence is geometric is to see whether the ratio between consecutive terms - the number I would multiply one term by to get the next - is constant.
By inspection, we see that the fourth answer choice satisfies that, as [tex]\frac{81}{27}=\frac{27}{9}=\frac{9}{3}=\frac{3}{1}=3.[/tex] Why not the first? We have [tex]2=\frac{12}{6}\ne\frac{6}{2}=3.[/tex]
The third choice is not a geometric sequence, but rather an arithmetic sequence, where the difference between consecutive terms is constant. Just to make sure that it isn't geometric, we compute [tex]\frac{14}{9}\ne\frac{9}{4}.[/tex]
The second sequence is not geometric (although it does eventually converge to 1, but not its corresponding series), as [tex]\frac{4}{3}=\frac{2/3}{1/2}\ne\frac{3/4}{2/3}=\frac{9}{8}.[/tex]
