Question

Which of the following is a geometric sequence?

Answer 1

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 $\frac{81}{27}=\frac{27}{9}=\frac{9}{3}=\frac{3}{1}=3.$ Why not the first? We have $2=\frac{12}{6}\ne\frac{6}{2}=3.$

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 $\frac{14}{9}\ne\frac{9}{4}.$

The second sequence is not geometric (although it does eventually converge to 1, but not its corresponding series), as $\frac{4}{3}=\frac{2/3}{1/2}\ne\frac{3/4}{2/3}=\frac{9}{8}.$