## Calculus with Applications (10th Edition)

$${S_5} = \frac{{1111}}{{72}}$$
\eqalign{ & {a_1} = 18,{a_2} = - 3,{a_3} = 1/2,{a_4} = - 1/12,... \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{{ - 1/12}}{{1/2}} = - \frac{1}{6} \cr & \frac{{{a_3}}}{{{a_2}}} = \frac{{1/2}}{{ - 3}} = - \frac{1}{6} \cr & \frac{{{a_2}}}{{{a_1}}} = \frac{{ - 3}}{{18}} = - \frac{1}{6} \cr & {\text{the ratio is constant}}{\text{, so the sequence is geometric with }}r = - \frac{1}{6} \cr & {\text{then the sum of the first }}n{\text{ terms}}{\text{ is given by}} \cr & {S_n} = \frac{{{a_1}\left( {{r^n} - 1} \right)}}{{r - 1}},{\text{ where }}r \ne 1 \cr & {\text{let }}n = 5,\,\,\,{a_1} = 12{\text{ and }}r = - \frac{1}{2} \cr & {S_5} = \frac{{\left( {18} \right)\left( {{{\left( { - \frac{1}{6}} \right)}^5} - 1} \right)}}{{ - \frac{1}{6} - 1}} \cr & {\text{simplifying}} \cr & {S_5} = \frac{{1111}}{{72}} \cr}