В геометрической прогрессии {a_n} a_9=15, a_11=135. Найдите a_10.

\[a_{9} = 15 \Longrightarrow a_{9} = b_{1}q^{8};\ \ \ \]

\[a_{11} = 135 \Longrightarrow a_{11} = b_{1}q^{10}\]

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

\[a_{9} \cdot a_{11} = b_{1}q^{8} \cdot b_{1}q^{10} =\]

\[= b_{1}^{2} \cdot q^{18} = \left( b_{1}q^{9} \right)^{2} = \left( a_{10} \right)^{2}\]

\[\left( a_{10} \right)^{2} = 15 \cdot 135 =\]

\[= 2025 \Longrightarrow a_{10} = 45\ \ \]

\[или\ \ a_{10} = - 45.\]

\[Ответ: - 45\ \ или\ 45.\ \]