Answer
$${S_5} \approx - 6.221$$
Work Step by Step
$$\eqalign{
& {\text{the sum of the first }}n{\text{ terms of a geometric sequence}}{\text{ is given by}} \cr
& {S_n} = \frac{{{a_1}\left( {{r^n} - 1} \right)}}{{r - 1}},{\text{ where }}r \ne 1 \cr
& {\text{let }}n = 5,\,\,\,{a_1} = - 2.772{\text{ and }}r = - 1.335 \cr
& {S_5} = \frac{{\left( { - 2.772} \right)\left( {{{\left( { - 1.335} \right)}^5} - 1} \right)}}{{ - 1.335 - 1}} \cr
& {\text{simplifying with a calculator}} \cr
& {S_5} \approx - 6.221 \cr} $$