Question

Como fazer fatorial com incógnita/equação?

a. $\frac{(x+1)!}{x!}$
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b. $x! + \frac{(x+1)!}{(x+3)!}$
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c. $\frac{(N+1)! - N!}{(N+1)!}$
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d. $\frac{(x+2)! + (x+1)!}{(x+3)!}$

Answer 1

Para questões desse tipo, uma das saídas é aplicarmos a definição de fatorial e buscarmos cancelamentos, de forma a simplificar as frações dadas:

$a)~\dfrac{(x+1)!}{x!}=\dfrac{(x+1)x!}{x!}=(x+1)\\\\\\ b)~x!+\dfrac{(x+1)!}{(x+3)!}=x!+\dfrac{(x+1)!}{(x+3)(x+2)(x+1)!}=x!+\dfrac{1}{(x+3)(x+2)}$


$c)~\dfrac{(n+1)!-n!}{(n+1)!}=\dfrac{(n+1)n!-n!}{(n+1)n!}=\dfrac{n![(n+1)-1]}{(n+1)n!}=\dfrac{n}{n+1}\\\\\\ d)~\dfrac{(x+2)!+(x+1)!}{(x+3)!}=\dfrac{(x+1)![(x+2)+1]}{(x+3)(x+2)(x+1)!}=\dfrac{x+3}{(x+3)(x+2)}=\\\\=\dfrac{1}{x+2}$