Question

Resolva a equação abaixo:

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Answer 1

$\sf     \Bigg( \frac{(n + 10)!}{(n + 8)!} \Bigg) = 30 \\ \\ \\ \sf     \frac{(n + 10) \cdot(n + 9) \cdot \cancel{(n + 8)!}}{ \cancel{(n + 8)!}} = 30 \\ \\ \\ \sf     (n + 10) \cdot (n + 9) = 30 \\ \\ \\ \sf     n \cdot n + 9n + 10n + 10 \cdot9 = 30 \\ \\ \\ \sf     n {}^{2} + 9n + 10n + 10 \cdot9 = 30 \\ \\ \\ \sf     n {}^{2} + 9n + 10n + 90 = 30 \\ \\ \\ \sf     n {}^{2} + 19n + 90 = 30 \\ \\ \\ \sf     n {}^{2} + 19n + 90 - 30 = 0 \\ \\ \\ \sf     n {}^{2} + 15n + 4n + 90 - 30 = 0 \\ \\ \\ \sf     n {}^{2} + 15n + 4n + 60 = 0 \\ \\ \\ \sf     n \cdot(n + 15) + 4n + 60 = 0 \\ \\ \\ \sf     n \cdot(n + 15) + 4 \cdot(n + 15) = 0 \\ \\ \\ \sf     (n + 15) \cdot(n + 4) = 0 \\ \\ \\ \sf     n + 15 = 0 \\ \sf     n + 4 = 0 \\ \\ \\ \sf     {\color{Blue}n = - 15} \\ \sf    {\color{Blue} n = - 4} \\ \\ \\ \sf     \frac{( - 15 + 10)!}{( - 15 + 8)!} = 30 \\ \\ \sf     n = 4 \\ \\ \\ \sf     \frac{ ( - 15 + 10)!}{( - 15 + 8)!} = 30 \\ \\ \sf     \frac{( - 4 + 10)!}{( - 4 + 8)!} = 30 \\ \\ \\ \sf     \frac{( - 5)!}{( - 15 + 8)!} = 30 \\ \\ \sf     \frac{( - 4 + 10)!}{( - 4 + 8)!} = 30 \\ \\\sf     \frac{6!}{( - 4 + 8)!} = 30 \\ \\ \sf     \frac{6!}{4!} = 30 \\ \\ \sf     \frac{6 \cdot5 \cdot \cancel{4!}}{ \cancel{4!}} = 30 \\ \\ \sf     6 \cdot5 = 30 \\ \\ \sf     {\color{Blue}30 = 30} \\ \\ \\ \sf     n \ne - 15 \\ {\color{Blue} \boxed{\sf n = - 4 {\color{lime}\longleftarrow Resposta}}}$

Att: Nerd1990