\[\boxed{\text{664}\text{\ (664)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Формула\ верна\ при\ n = 1:\ \ \frac{1}{n(n + 1)} = \frac{1}{2} = \frac{1}{1 + 1}.\]
\[Допустим,\ что\ при\ n = k,\ формула\ тоже\ верна \Longrightarrow то\ есть,\]
\[\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{k(k + 1)} = \frac{k}{k + 1}.\]
\[Докажем,\ что\ формула\ справедлива\ для\ n = k + 1:\]
\[\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \ldots + \frac{1}{k(k + 1)} + \frac{1}{(k + 1)(k + 2)} =\]
\[= \frac{1}{(k + 1)} + \frac{1}{(k + 1)(k + 2)} = \frac{1}{(k + 1)} \cdot \left( k + \frac{1}{k + 2} \right) =\]
\[= \frac{1}{(k + 1)} \cdot \frac{k² + 2k + 1}{k + 2} = \frac{(k + 1)²}{(k + 1)(k + 2)} = \frac{k + 1}{k + 2} \Longrightarrow ч.т.д.\]
\[\boxed{\text{664.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[\textbf{а)}\ S_{20} = 1000,\ \ S_{40} = 10000;\]
\[S_{20} = \frac{2a_{1} + 19d}{2} \cdot 20 =\]
\[= 10 \cdot \left( 2a_{1} + 19d \right) = 1000 \Longrightarrow\]
\[2a_{1} + 19d = 100;\]
\[S_{40} = \frac{2a_{1} + 39d}{2} \cdot 40 =\]
\[= 20 \cdot \left( 2a_{1} + 39d \right) = 1000 \Longrightarrow\]
\[2a_{1} + 39d = 500;\]
\[\left\{ \begin{matrix} 2a_{1} + 19d = 100 \\ 2a_{1} + 39d = 500 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 20d = 400\ \ \ \ \ \ \ \\ a_{1} = \frac{100 - 19d}{2} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} d = 20\ \ \ \ \ \ \ \ \\ a_{1} = - 140 \\ \end{matrix} \right.\ \Longrightarrow\]
\[a_{50} = a_{1} + d(n - 1) =\]
\[= - 140 + 20 \cdot 49 = 840;\]
\[\textbf{б)}\ S_{5} = 0,5;\ \ \ \ \ S_{15} = - 81;\]
\[S_{5} = \frac{2a_{1} + 4d}{2} \cdot 5 =\]
\[= 5 \cdot \left( a_{1} + 2d \right) = 0,5 \Longrightarrow\]
\[\Longrightarrow a_{1} + 2d = 0,1;\]
\[S_{15} = \frac{2a_{1} + 14d}{2} \cdot 15 =\]
\[= 15 \cdot \left( a_{1} + 7d \right) = - 81 \Longrightarrow\]
\[\Longrightarrow a_{1} + 7d = - 5,4;\]
\[\left\{ \begin{matrix} a_{1} + 2d = 0,1\ \ \\ a_{1} + 7d = - 5,4 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 5d = - 5,5\ \ \ \ \ \\ a_{1} = 0,1 - 2d \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} d = - 1,1 \\ a_{1} = 2,3. \\ \end{matrix} \right.\ \]
\[a_{50} = a_{1} + 49d =\]
\[= 2,3 - 1,1 \cdot 49 = - 51,6.\]