## Calculus with Applications (10th Edition)

$${a_4} = - 250,\,\,\,\,\,{a_n} = 2{\left( { - 5} \right)^{n - 1}}{\text{ and }}{S_5} = 1042$$
\eqalign{ & {a_1} = 2,\,\,\,\,r = - 5 \cr & {\text{The general term of a geometric sequence is }}{a_n} = {a_1}{r^{n - 1}}.{\text{ }} \cr & {\text{substituting }}{a_1} = 2,\,\,\,\,r = - 5 \cr & {a_n} = 2{\left( { - 5} \right)^{n - 1}} \cr & \cr & {\text{find }}{a_4},{\text{ substitute }}n = 4{\text{ into the general term formula}} \cr & {a_4} = 2{\left( { - 5} \right)^{4 - 1}} \cr & {a_4} = 2\left( { - 125} \right) \cr & {a_4} = - 250 \cr & \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{For the first five terms we have}} \cr & {S_5} = \frac{{\left( 2 \right)\left( {{{\left( { - 5} \right)}^5} - 1} \right)}}{{ - 5 - 1}} \cr & {S_5} = \frac{{\left( 2 \right)\left( { - 3125 - 1} \right)}}{{ - 6}} \cr & {S_5} = 1042 \cr & \cr & {\text{then}} \cr & {a_4} = - 250,\,\,\,\,\,{a_n} = 2{\left( { - 5} \right)^{n - 1}}{\text{ and }}{S_5} = 1042 \cr}