В геометрической прогрессии {a_n} a_14=24, a_16=54. Найдите a_15.

\[a_{14} = 24 \Longrightarrow a_{14} = b_{1}q^{13}\]

\[a_{16} = 54 \Longrightarrow a_{16} = b_{1}q^{15}\]

\[a_{15} = b_{1}q^{14}\]

\[a_{14} \cdot a_{16} = b_{1}q^{13} \cdot b_{1}q^{15} =\]

\[= b_{1}^{2} \cdot q^{28} = \left( b_{1}q^{14} \right)^{2} = \left( a_{15} \right)^{2}\]

\[\left( a_{15} \right)^{2} = 24 \cdot 54 = 1296 \Longrightarrow\]

\[\Longrightarrow a_{15} = 36\ \ или\ \ a_{15} = - 36.\]

\[Ответ:\ - 36\ \ или\ \ 36.\]