Respostas

  • Usuário do Brainly
2013-06-01T13:52:47-03:00

Temos que:

 

\dfrac{(\text{n}+1)!-\text{n}!}{(\text{n}+1)!+\text{n}!}=\dfrac{2}{3}

 

Observe que:

 

(\text{n}+1)!=(\text{n}+1)\cdot\text{n}!

 

Logo:

 

\dfrac{(\text{n}+1)\cdot\text{n}!-\text{n}!}{(\text{n}+1)\cdot\text{n}!+\text{n}!}=\dfrac{2}{3}

 

Fatorando o numerador e o denominador, segue que:

 

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

 

Donde, obtemos:

 

\dfrac{(\text{n}+1)-1}{(\text{n}+1)+1}=\dfrac{2}{3}

 

\dfrac{\text{n}}{\text{n}+2}=\dfrac{2}{3}

 

Logo:

 

3\text{n}=2\text{n}+4

 

\text{n}=4