ГДЗ по алгебре 9 класс Макарычев Задание 1071

\[\boxed{\text{1071\ (1071).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[x_{n} = \frac{1}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}},\ \ \]

\[S_{n} = \frac{n}{\sqrt{a_{1}} + \sqrt{a_{n + 1}}},\]

\[x_{n} = \frac{1}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}} =\]

\[= \frac{\sqrt{a_{n + 1}} - \sqrt{a_{n}}}{\left( \sqrt{a_{n}} + \sqrt{a_{n + 1}} \right)\left( \sqrt{a_{n + 1}} - \sqrt{a_{n}} \right)} =\]

\[= \frac{\sqrt{a_{n + 1}} - \sqrt{a_{n}}}{a_{n + 1} - a_{n}} =\]

\[= \frac{\sqrt{a_{n + 1}} - \sqrt{a_{n}}}{d}.\]

\[x_{1} = \frac{\sqrt{a_{2}} - \sqrt{a_{1}}}{d},\]

\[x_{2} = \frac{\sqrt{a_{3}} - \sqrt{a_{2}}}{d},\]

\[x_{n - 1} = \frac{\sqrt{a_{n}} - \sqrt{a_{n - 1}}}{d},\]

\[x_{n} = \frac{\sqrt{a_{n + 1}} - \sqrt{a_{n}}}{d}.\]

\[S_{n} = x_{1} + x_{2} + \ldots + x_{n - 1} + x_{n} =\]

\[= \frac{1}{d} \cdot \left( \sqrt{a_{n + 1}} - \sqrt{a_{1}} \right) =\]

\[= \frac{\left( \sqrt{a_{n + 1}} - \sqrt{a_{1}} \right)\left( \sqrt{a_{n + 1}} + \sqrt{a_{1}} \right)}{d\left( \sqrt{a_{n + 1}} + \sqrt{a_{1}} \right)} =\]

\[= \frac{a_{n + 1} - a_{1}}{d\left( \sqrt{a_{n + 1}} + \sqrt{a_{1}} \right)} =\]

\[= \frac{a_{1} + nd - a_{1}}{d\left( \sqrt{a_{n + 1}} + \sqrt{a_{1}} \right)} =\]

\[= \frac{n}{\sqrt{a_{1}} + \sqrt{a_{n + 1}}} \Longrightarrow ч.т.д.\]