\(\begin{array}{l}\left\{ \begin{array}{l}{u_1} + {u_3} + {u_5} = 182\\{u_1} + {u_5} = 164\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{u_1} + {u_1}{q^2} + {u_1}{q^4} = 182\\{u_1} + {u_1}{q^4} = 164\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}{u_1}\left( {1 + {q^2} + {q^4}} \right) = 182\\{u_1}\left( {1 + {q^4}} \right) = 164\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\dfrac{{1 + {q^2} + {q^4}}}{{1 + {q^4}}} = \dfrac{{182}}{{164}}\\{u_1}\left( {1 + {q^4}} \right) = 164\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}164 + 164{q^2} + 164{q^4} = 182 + 182{q^4}\\{u_1}\left( {1 + {q^4}} \right) = 164\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}18{q^4} - 164{q^2} + 18 = 0\\{u_1}\left( {1 + {q^4}} \right) = 164\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\left[ \begin{array}{l}{q^2} = 9\\{q^2} = \dfrac{1}{9}\end{array} \right.\\{u_1}\left( {1 + {q^4}} \right) = 164\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}q = 3,{u_1} = 2\\q = - 3,{u_1} = 2(loai)\\q = \dfrac{1}{3},{u_1} = 162\\q = - \dfrac{1}{3},{u_1} = 162(loai)\end{array} \right.\end{array}\)
TH1: \(q = 3,{u_1} = 2 \Rightarrow {u_8} = {u_1}{q^7} = {2.3^7}\)
\({S_{10}} = \dfrac{{{u_1}\left( {{q^{10}} - 1} \right)}}{{q - 1}} = \dfrac{{2\left( {{3^{10}} - 1} \right)}}{{3 - 1}} = {3^{10}} - 1\)
TH2: \(q = \dfrac{1}{3},{u_1} = 162 \Rightarrow {u_8} = {u_1}{q^7} = 162.{\left( {\dfrac{1}{3}} \right)^7} = \dfrac{2}{{27}}\)
\({S_{10}} = \dfrac{{{u_1}\left( {{q^{10}} - 1} \right)}}{{q - 1}} = \dfrac{{162\left( {{{\left( {\dfrac{1}{3}} \right)}^{10}} - 1} \right)}}{{\dfrac{1}{3} - 1}} = - 243\left( {{{\left( {\dfrac{1}{3}} \right)}^{10}} - 1} \right)\)
