\[\boxed{\text{681}\text{\ (681)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Дано:\left( y_{n} \right) - арифметическая\ прогрессия.\]
\[Доказать:\]
\[\textbf{а)}\ y_{2} + y_{7} = y_{4} + y_{5}\]
\[y_{2} = y_{1} + d,\]
\[y_{7} = y_{1} + 6d,\]
\[y_{4} = y_{1} + 3d\]
\[y_{5} = y_{1} + 4d\]
\[y_{1} + d + y_{1} + 6d = y_{1} + 3d + y_{1} + 4d\]
\[2y_{1} + 7d = 2y_{1} + 7d \Longrightarrow верно.\]
\[\textbf{б)}\ y_{n - 5} + y_{n + 10} = y_{n} + y_{n + 5},\ \ где\ \ n > 5.\]
\[y_{n + 10} = y_{1} + d(n + 9)\]
\[y_{n - 5} = y_{1} + d(n - 6)\]
\[y_{n} = y_{1} + d(n - 1)\]
\[y_{n + 5} = y_{1} + d(n + 4)\]
\[y_{1} + d(n - 6) + y_{1} + d(n + 9) = y_{1} + d(n - 1) + y_{1} + d(n + 4)\]
\[2y_{1} + d(2n + 3) = 2y_{1} + d(2n + 3) \Longrightarrow ч.т.д.\]