Найдите первый член и разность арифметической прогрессии (an), если: a2+a6=24 и a2*a3=54.

Ответ:

\[\left\{ \begin{matrix} a_{2} + a_{6} = 24 \\ a_{2} \cdot a_{3} = 54\ \ \\ \end{matrix}\text{\ \ \ } \right.\ \]

\[\left\{ \begin{matrix} a_{1} + d + a_{1} + 5d = 24 \\ \left( a_{1} + d \right)\left( a_{1} + 2d \right) = 54 \\ \end{matrix}\text{\ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 2a_{1} + 6d = 24\ \ \ \ \ \ \ \ \ |\ :2 \\ \left( a_{1} + d \right)\left( a_{1} + 2d \right) = 54 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} a_{1} = 12 - 3d\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (12 - 2d)(12 - d) = 54 \\ \end{matrix} \right.\ \]

\[d^{2} - 18d + 45 = 0\]

\[d_{1} + d_{2} = 18,\ \ d_{1} \cdot d_{2} = 45\ \ \]

\[d = 3,\ \ d = 15\]

\[\left\{ \begin{matrix} a_{1} = 3 \\ d = 3\ \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} a = - 33 \\ d = 5\ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:a = 3,\ \ \ d = 3\ или\ \ \ \]

\[a = - 33,\ \ \ d = 15.\]