\[\boxed{\text{656}\text{\ (656)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[b_{2} = 6,\ \ b_{4} = 54,\ \ q > 0;\]
\[b_{4} = b_{2} \cdot q^{2}\]
\[q^{2} = \frac{b_{4}}{b_{2}}\]
\[q^{2} = \frac{54}{6} = 9\]
\[q = 3\]
\[q = - 3 \Longrightarrow не\ подходит\ по\ условию.\]
\[\Longrightarrow b_{2} = b_{1} \cdot q \Longrightarrow b_{1} = \frac{b_{2}}{q} \Longrightarrow b_{1} = \frac{6}{3} = 2.\]
\[S_{7} = \frac{b_{1} \cdot (q^{7} - 1)}{q - 1} = \frac{2 \cdot (3^{7} - 1)}{3 - 1} = 2186.\]