В геометрической прогрессии {a_n} a_10=27, a_12=108. Найдите a_11.

\[a_{10} = 27 \Longrightarrow a_{10} = b_{1}q^{9};\]

\[a_{12} = 108 \Longrightarrow a_{12} = b_{1}q^{11}\]

\[a_{11} = b_{1}q^{10}\]

\[a_{10} \cdot a_{12} = b_{1}q^{9} \cdot b_{1}q^{11} =\]

\[= b_{1}^{2} \cdot q^{20} = \left( b_{1} \cdot q^{10} \right)^{2} = \left( a_{11} \right)^{2}\]

\[\left( a_{11} \right)^{2} = 27 \cdot 108 = 2916 \Longrightarrow\]

\[\Longrightarrow a_{11} = 54\ \ или\ \ a_{11} = - 54\]

\[Ответ:\ - 54\ или\ 54.\]