ГДЗ по алгебре 9 класс Макарычев Задание 840

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\[\textbf{а)}\ \frac{(n + 1)!}{(n - 1)!} = 42\]

\[\frac{(n - 1)! \cdot n(n + 1)}{(n - 1)!} = 42\]

\[n(n + 1) = 42\]

\[n^{2} + n - 42 = 0,\ \ n > 0\]

\[По\ теореме\ Виета:n_{1} = - 7,\ \ \]

\[n_{2} = 6.\]

\[n_{2} = - 7 \Longrightarrow не\ подходит.\]

\[Ответ:n = 6.\]

\[\textbf{б)}\ \frac{(n + 1)! - n!}{(n + 1)!} = \frac{5}{6}\ \]

\[\frac{n!(n + 1) - n!}{n!(n + 1)} = \frac{5}{6}\]

\[\frac{n!\left( (n + 1) - 1 \right)}{n!(n + 1)} = \frac{5}{6}\]

\[\frac{n}{n + 1} = \frac{5}{6}\]

\[n = 5\]

\[Ответ:n = 5.\]