Решебник по алгебре 9 класс Мерзляк Задание 806

\[\boxed{\mathbf{806\ (806).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ 11 + 19 + 27 + \ldots +\]

\[+ (8n + 3) = 470\]

\[a_{1} = 11,\ \ d = 19 - 11 = 8,\ \]

\[\ a_{n} = 8n + 3,\ \ S_{n} = 470\]

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

\[470 = \frac{11 + 8n + 3}{2} \cdot n\ \ \]

\[470 = \frac{8n + 14}{2} \cdot n\]

\[470 = (4n + 7) \cdot n\]

\[4n^{2} + 7n - 470 = 0,\ \ \]

\[D = 49 + 7520 = 7569\]

\[n = \frac{- 7 - 87}{8} < 0\ \ \]

\[n = \frac{- 7 + 87}{8} = 10\]

\[Ответ:n = 10.\]

\[2)\ 1 + 5 + 9 + \ldots + x = 630\]

\[a_{1} = 1,\ \ d = 5 - 1 = 4,\]

\[\text{\ \ }a_{n} = x,\ \ S_{n} = 630\]

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

\[630 = \frac{2 + 4 \cdot (n - 1)}{2} \cdot n\]

\[630 = \left( 1 + 2 \cdot (n - 1) \right) \cdot n\]

\[630 = (1 + 2n - 2) \cdot n\ \]

\[630 = (2n - 1) \cdot n\]

\[2n^{2} - n - 630 = 0\ \]

\[D = 1 + 5040 = 5041\]

\[n = \frac{1 - 71}{4} < 0\ \]

\[n = \frac{1 + 71}{4} = 18\]

\[a_{n} = a_{1} + d(n - 1) = x\ \ \]

\[x = 1 + 4 \cdot (18 - 1) =\]

\[= 1 + 4 \cdot 17 = 69\]

\[Ответ:x = 69.\]