Question

Dada a equação: (n-1)! + n!/n+1=120

Qual o valor de n?​

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{\dfrac{(n - 1)! + n!}{(n + 1)!} = 120}$

$\mathsf{\dfrac{(n - 1)! + n.(n - 1)!}{(n + 1).n.(n - 1)!} = 120}$

$\mathsf{\dfrac{1 + n}{n(n + 1)} = 120}$

$\mathsf{\dfrac{1 + n}{n^2 + n} = 120}$

$\mathsf{120n^2 + 120n = 1 + n}$

$\mathsf{120n^2 + 119n - 1 = 0}$

$\mathsf{\Delta = b^2 - 4.a.c}$

$\mathsf{\Delta = (119)^2 - 4.120.(-1)}$

$\mathsf{\Delta = 14.161 + 480}$

$\mathsf{\Delta = 14.641}$

$\mathsf{n = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{-119 \pm \sqrt{14.641}}{240} \rightarrow \begin{cases}\mathsf{n' = \dfrac{-119 + 121}{240} = \dfrac{2}{240} = \dfrac{1}{120}}\\\\\mathsf{n'' = \dfrac{-119 - 121}{240} = \dfrac{-240}{240} = -1}\end{cases}}$

$\boxed{\boxed{\mathsf{S = \{\:\dfrac{1}{120}\:\}}}}\leftarrow\textsf{{\'u}nico valor poss{\'i}vel para n}$