Решебник по алгебре 9 класс Макарычев Задание 694

 

\[\boxed{\text{694}\text{\ (694)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ \frac{x \cdot x^{2} \cdot x^{3} \cdot .. \cdot x^{n}}{x \cdot x^{3} \cdot x^{5} \cdot .. \cdot x^{2n - 1}}\]

\[1)\ a_{1} = 1,\ \ a_{2} = 2,\ \ d = 1,\]

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

\[2)\ \ 1 + 3 + 5 + .. + 2n - 1,\]

\[a_{1} = 1,\ \ a_{2} = 3,\ \ d = 2,\]

\[a_{k} = a_{1} + d(k - 1) = 1 + 2k - 2 = 2k - 1,\]

\[k = n,\]

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

\[\frac{x \cdot x^{2} \cdot x^{3} \cdot .. \cdot x^{n}}{x \cdot x^{3} \cdot x^{5} \cdot .. \cdot x^{2n - 1}} = \frac{x^{\frac{2n² + n}{2}}}{x^{n²}} = x^{\frac{n² + n - 2n²}{2}} = x^{\frac{n - n²}{n}};\]

\[\textbf{б)}\ \frac{x^{2} \cdot x^{4} \cdot x^{6}.. \cdot x^{2n}}{x \cdot x^{2} \cdot x^{3} \cdot .. \cdot x^{n}} = \frac{x^{2 + 4 + 6 + .. + 2n}}{x^{1 + 2 + 3 + .. + n}} = \frac{\left( x^{1 + 2 + 3 + .. + n} \right)^{2}}{x^{1 + 2 + 3 + .. + n}} = x^{1 + 2 + 3 + .. + n} =\]

\[= x^{\frac{n(n + 1)}{2}.}\]

\[\boxed{\text{694.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]

\[\textbf{а)}\ a_{2} = - 6,\ \ a_{3} = - 2,\ \ \]

\[a_{15} = ?\]

\[\left\{ \begin{matrix} a_{2} = a_{1} + d\ \ \\ a_{3} = a_{1} + 2d \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} a_{1} + d = - 6\ \ \\ a_{1} + 2d = - 2 \\ \end{matrix} \right.\ \ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} a_{1} + d = - 6\ \ \ \ \ \ \ \ \ \ \ \ \ \\ d = - 2 - ( - 6) = 4 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} d = 4\ \ \ \ \ \ \ \ \\ a_{1} = - 10, \\ \end{matrix} \right.\ \]

\[a_{15} = a_{1} + 14d =\]

\[= - 10 + 14 \cdot 4 = 46.\]

\[\textbf{б)}\ x_{2} = - 2,4;\ \ \ d = 1,2;\ \ \ S_{10} = ?\]

\[x_{2} = x_{1} + d\]

\[x_{1} = x_{2} - d = - 24 - 12 = - 36.\]

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

\[= \frac{- 7,2 + 1,2 \cdot 9}{2} \cdot 10 = 18.\]