\[\boxed{\text{712\ (712).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[S_{5} = \frac{11}{64},\ \ S_{10} - S_{5} = - 5\frac{1}{2},\]
\[S_{5} = x_{1} \cdot \frac{q^{5} - 1}{q - 1},\ \ \]
\[S_{10} = x_{1} \cdot \frac{q^{10} - 1}{q - 1},\]
\[S_{10} - S_{5} =\]
\[= \frac{x_{1}}{q - 1} \cdot \left( q^{10} - 1 - q^{5} + 1 \right) =\]
\[= \frac{x_{1} \cdot q^{5} \cdot \left( q^{5} - 1 \right)}{q - 1} = q^{6} \cdot S_{5},\]
\[q^{5} = \frac{S_{10} - S_{5}}{S_{5}} = - 5\frac{1}{2} \cdot \frac{64}{11} = - 32,\]
\[q = - 2,\]
\[S_{15} - S_{10} =\]
\[= x_{1} \cdot \frac{q^{15} - 1}{q - 1} - x_{1} \cdot \ \frac{q^{10} - 1}{q - 1} =\]
\[= \frac{x_{1}}{q - 1} \cdot \left( q^{15} - 1 - q^{10} + 1 \right) =\]
\[= \frac{x_{1} \cdot q^{10} \cdot \left( q^{5} - 1 \right)}{q - 1} = q^{10} \cdot S_{5} =\]
\[= ( - 2)^{10} \cdot \frac{11}{64} = 176.\]