\[b_{1} = \frac{1}{8};\ \ \ q = 2:\]
\[b_{6} = \frac{1}{8} \cdot q^{5} = \frac{1}{8} \cdot 32 = 4.\]
\[S_{6} = \frac{\frac{1}{8} \cdot \left( q^{6} - 1 \right)}{q - 1} = \frac{\frac{1}{8} \cdot (64 - 1)}{2 - 1} =\]
\[= \frac{1}{8} \cdot 63 = \frac{63}{8} = 7\frac{7}{8}.\]
\[Ответ:7\frac{7}{8}.\]