Question

(t
tikote
$(n + 4)! \\ (n + 2)! + (n + 3)!$
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Answer 1

Explicação passo-a-passo:

$\text{S}=\dfrac{(\text{n}+4)!}{(\text{n}+2)!+(\text{n}+3)!}$

Veja que

$\bullet~(\text{n}+4)!=(\text{n}+4)\cdot(\text{n}+3)\cdot(\text{n}+2)!$

$\bullet~(\text{n}+3)!=(\text{n}+3)\cdot(\text{n}+2)!$

Substituindo:

$\text{S}=\dfrac{(\text{n}+4)\cdot(\text{n}+3)\cdot(\text{n}+2)!}{(\text{n}+2)!+(\text{n}+3)\cdot(\text{n}+2)!}$

Colocando $(\text{n}+2)!$ em evidência:

$\text{S}=\dfrac{(\text{n}+2)!\cdot(\text{n}+4)\cdot(\text{n}+3)}{(\text{n}+2)!\cdot[1+(\text{n}+3)]}$

$\text{S}=\dfrac{(\text{n}+4)\cdot(\text{n}+3)}{1+(\text{n}+3)}$

$\text{S}=\dfrac{(\text{n}+4)\cdot(\text{n}+3)}{(\text{n}+4)}$

$\text{S}=\text{n}+3$