Решебник по алгебре 9 класс Мерзляк Задание 1032

\[\boxed{\mathbf{1032\ (1032).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[S_{3} = 3,\ \ S_{4} = 16,\ \ \]

\[S_{n} = 220\]

\[S_{4} = S_{3} + a_{4},\ \ a_{4} = S_{4} -\]

\[- S_{3} = 16 - 3 = 13\]

\[S_{4} = \frac{a_{1} + a_{4}}{2} \cdot 4 =\]

\[= \left( a_{1} + 13 \right) \cdot 2 = 16\]

\[a_{1} + 13 = 8 \Longrightarrow \ \ a_{1} = - 5\]

\[a_{4} = a_{1} + 3d \Longrightarrow \ \ 3d = a_{4} - a_{1},\ \ \]

\[d = \frac{a_{4} - a_{1}}{3} = \frac{13 + 5}{3} = \frac{18}{3} = 6\]

\[S_{n} = \frac{2a_{1} + d(n - 1)}{2} \cdot n =\]

\[= \frac{- 10 + 6 \cdot (n - 1)}{2} \cdot n =\]

\[= \left( - 5 + 3 \cdot (n - 1) \right) \cdot n\]

\[- 5n + 3n(n - 1) = 220\]

\[- 5n + 3n^{2} - 3n - 220 = 0\]

\[3n^{2} - 8n - 220 = 0\]

\[D = 64 + 2640 = 2704\]

\[n = \frac{8 - 52}{6} < 0\]

\[n = \frac{8 + 52}{6} = 10\]

\[Ответ:n = 10.\]