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МатематикаРешебник по алгебре 9 класс Мерзляк Задание 737
Задание 737
\[\boxed{\mathbf{737\ (737).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]
\[a_{n} = a_{1} + d(n - 1)\]
\[1)\left\{ \begin{matrix} a_{3} + a_{7} = 30 \\ a_{6} + a_{16} = 60 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]
\[\ \left\{ \begin{matrix} a_{1} + 2d + a_{1} + 6d = 30 \\ a_{1} + 5d + a_{1} + 15d = 60 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]
\[\text{\ \ \ \ \ }\left\{ \begin{matrix} 2a_{1} + 8d = 30 \\ 2a_{1} + 20d = 60 \\ \end{matrix} \right.\ \ -\]
\[\left\{ \begin{matrix} - 12d = - 30 \\ a_{1} = \frac{30 - 8d}{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]
\[\text{\ \ \ \ }\left\{ \begin{matrix} d = 2,5 \\ a_{1} = \frac{30 - 20}{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} d = 2,5 \\ a_{1} = 5\ \ \\ \end{matrix} \right.\ \]
\[Ответ:\ a_{1} = 5;\ \ \ d = 2,5.\]
\[2)\ \left\{ \begin{matrix} a_{n} + a_{10} = 36 \\ a_{5} \cdot a_{11} = 340 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} a_{1} + 3d + a_{1} + 9d = 36\ \ |\ :2 \\ \left( a_{1} + 4d \right)\left( a_{1} + 10d \right) = 340\ \ \\ \end{matrix} \right.\ \text{\ \ }\ \]
\[\ \left\{ \begin{matrix} a_{1} + 6d = 18\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \left( a_{1} + 4d \right)\left( a_{1} + 10d \right) = 340 \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} a_{1} = 18 - 6d\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (18 - 6d + 4d)(18 - 6d + 10d) = 340 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} a_{1} = 18 - 6d\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (18 - 2d)(18 + 4d) = 340\ \ \ |\ :4 \\ \end{matrix} \right.\ \]
\[(9 - d)(9 + 2d) = 85\]
\[81 + 18d - 9d - 2d^{2} = 85\]
\[- 2d^{2} + 9d - 4 = 0\]
\[2d^{2} - 9d + 4 = 0\]
\[D = 81 - 32 = 49\]
\[d_{1} = \frac{9 + 7}{4} = 4;\ \ \ \ \ \]
\[\ d_{2} = \frac{9 - 7}{4} = 0,5\]
\[a_{1} = 18 - 6 \cdot 4 = 18 - 24 = - 6;\ \]
\[\text{\ \ \ }a_{1} = 18 - 6 \cdot 0,5 = 15\]
\[Ответ:d = 4;\ a_{1} = - 6\ \ \ или\ \ \ \]
\[d = 0,5;\ a_{1} = 15.\]
