Answer
$${S_4} = \frac{{13}}{{36}}$$
Work Step by Step
$$\eqalign{
& \frac{3}{4},\,\, - \frac{1}{2},\,\,\frac{1}{3},... \cr
& {\text{Let }}{a_1} = \frac{3}{4},\,\,\,{a_2} = - \frac{1}{2},{\text{ }}{a_3} = \frac{1}{3} \cr
& r = \frac{{{a_{n + 1}}}}{{{a_n}}} \cr
& r = \frac{{ - 1/2}}{{3/4}} = - \frac{2}{3} \cr
& {\text{Determine }}{S_4}{\text{ using }}{S_n} = \frac{{{a_1}\left( {1 - {r^n}} \right)}}{{1 - r}}\,\,\,\left( {{\text{where }}r \ne 1} \right) \cr
& {\text{Let }}n = 4 \cr
& {S_4} = \frac{{{a_1}\left( {1 - {r^4}} \right)}}{{1 - r}} \cr
& {S_4} = \frac{{\left( {3/4} \right)\left( {1 - {{\left( { - 2/3} \right)}^4}} \right)}}{{1 - \left( { - 2/3} \right)}} \cr
& {\text{Simplifying}} \cr
& {S_4} = \frac{{13}}{{36}} \cr} $$
