Answer
$$r = - \frac{2}{3}{\text{ and }}{a_n} = - \frac{5}{8}{\left( { - \frac{2}{3}} \right)^{n - 1}}$$
Work Step by Step
$$\eqalign{
& {a_1} = - \frac{5}{8},{a_2} = \frac{5}{{12}},{a_3} = - \frac{5}{{18}},{a_4} = \frac{5}{{27}},... \cr
& {\text{to verify that the sequence is geometric}}{\text{, divide each term except the first by }} \cr
& {\text{the preceding term}} \cr
& \frac{{{a_4}}}{{{a_3}}} = \frac{{5/27}}{{5/18}} = - \frac{2}{3} \cr
& \frac{{{a_3}}}{{{a_2}}} = \frac{{ - 5/18}}{{5/12}} = - \frac{2}{3} \cr
& \frac{{{a_2}}}{{{a_1}}} = \frac{{5/12}}{{ - 5/8}} = - \frac{2}{3} \cr
& {\text{the ratio is constant}}{\text{, so the sequence is geometric with }}r = - \frac{2}{3} \cr
& {\text{the general term of a geometric sequence is }}{a_n} = {a_1}{r^{n - 1}} \cr
& {\text{then}} \cr
& {a_n} = - \frac{5}{8}{\left( { - \frac{2}{3}} \right)^{n - 1}} \cr} $$
