В геометрической прогрессии {a_n} a_13=18, a_15=72. Найдите a_14.

\[a_{13} = 18 \Longrightarrow a_{13} = a_{1}q^{12};\]

\[a_{15} = 72 \Longrightarrow a_{15} = a_{1}q^{14}.\]

\[a_{14} = a_{1}q^{13}\]

\[a_{13} \cdot a_{15} = a_{1}q^{12} \cdot a_{1}q^{14} =\]

\[= a_{1}^{2} \cdot q^{26} = \left( a_{1}q^{13} \right)^{2} = \left( a_{14} \right)^{2}\]

\[\left( a_{14} \right)^{2} = 18 \cdot 72 = 1296 \Longrightarrow\]

\[\Longrightarrow a_{14} = 36\ \ \ или\ \ a_{14} = - 36.\]

\[Ответ:\ - 36\ \ или\ 36.\]