a_n – убывающая арифметическая прогрессия. a_1*a_3=-5; S_3=6. Найдите S_7.

\[a_{1} \cdot a_{3} = - 5;\ \ S_{3} = 6;\]

\[a_{3} = a_{1} + 2d;\]

\[S_{3} = \frac{a_{1} + a_{1} + 2d}{2} \cdot 3 = \frac{2a_{1} + 2d}{2} \cdot 3 = \left( a_{1} + d \right) \cdot 3\]

\[\left( a_{1} + d \right) \cdot 3 = 6\]

\[a_{1} + d = 2;\]

\[d = 2 - a_{1};\]

\[a_{1}\left( a_{1} + 2d \right) = - 5\]

\[a_{1}\left( a_{1} + 2\left( 2 - a_{1} \right) \right) = - 5\]

\[a_{1}\left( a_{1} + 4 - 2a_{1} \right) = - 5\]

\[- a_{1}^{2} + 4a_{1} + 5 = 0\]

\[a_{1}^{2} - 4a_{1} - 5 = 0\]

\[a_{1} = - 1;\ \ a_{1} = 5;\]

\[d_{1} = 2 + 1 = 3\ (не\ подходит,\ \]

\[так\ как\ прогрессия\ убывающая);\]

\[d = 2 - 5 = - 3;a_{1} = 5.\]

\[S_{7} = \frac{2a_{1} + 6d}{2} \cdot 7 =\]

\[= \frac{2 \cdot 5 + 6 \cdot ( - 3)}{2} \cdot 7 =\]

\[= \frac{10 - 18}{2} \cdot 7 = - \frac{8}{2} \cdot 7 = - 28.\]

\[Ответ:\ - 28.\]