Сумма восьмого и шестого членов арифметической прогрессии равна 16, а произведение второго и двенадцатого равно -36. Найдите разность и первый член прогрессии.

\[a_{8} + a_{6} = 16\]

\[a_{1} + 7d + a_{1} + 5d = 16\]

\[2a_{1} + 12d = 16\]

\[a_{1} + 6d = 8\]

\[a_{1} = 8 - 6d;\]

\[a_{2} \cdot a_{12} = - 36\]

\[\left( a_{1} + d \right)\left( a_{1} + 11d \right) = - 36\]

\[(8 - 6d + d)(8 - 6d + 11d) = - 36\]

\[(8 - 5d)(8 + 5d) = - 36\]

\[64 - 25d^{2} = - 36\]

\[- 25d^{2} = - 100\]

\[d^{2} = 4\]

\[d = \pm 2.\]

\[d = - 2:\ \]

\[a_{1} = 8 - 6d = 8 + 12 = 20;\]

\[d = 2:\]

\[a_{1} = 8 - 6d = 8 - 12 = - 4.\]

\[Ответ:d = - 2;\ \ a_{1} = 20\ \]

\[или\ d = 2;\ \ a_{1} = - 4.\]