\[\left\{ \begin{matrix} b_{4} - b_{2} = 30 \\ b_{4} - b_{3} = 24 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} b_{1}q^{3} - b_{1}q = 30 \\ b_{1}q^{3} - b_{1}q^{2} = 24 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]
\[\left\{ \begin{matrix} b_{1}q\left( q^{2} - 1 \right) = 30 \\ b_{1}q^{2}(q - 1) = 24 \\ \end{matrix} \right.\ \ \ \ |\ :\]
\[\frac{q + 1}{q} = \frac{5}{4}\]
\[4q + 4 = 5q\]
\[q = 4.\]
\[b_{1} = \frac{30}{q\left( q^{2} - 1 \right)} = \frac{30}{4 \cdot 15} =\]
\[= \frac{1}{2} = 0,5.\]
\[S_{5} = \frac{\frac{1}{2} \cdot \left( 4^{5} - 1 \right)}{4 - 1} =\]
\[= \frac{1023}{6} = 170,5.\]
\[Ответ:170,5.\]