ГДЗ по алгебре 9 класс Макарычев Задание 694

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

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

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

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

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

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

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

\[k = n,\]

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

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

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

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