Question

1) Simplifique a expressão
[(n+2)! - (n+1)!] .n! / [(n+1)!]², n ∈ N.

Answer 1

Olá Maslip,


Organizando a equação:


$\mathsf{\dfrac{[(n+2)! - (n+1)!]\cdot n!} {[(n+1)!]^2}~~~~~~~~~~~~~~~ ~\forall ~n\in \mathbb{N}}\\\\=\\\\\mathsf{\dfrac{[(n+2)\cdot(n+1)!-(n+1)!]\cdot n!}{(n+1)!\cdot(n+1)!}}\\\\=\\\\\mathsf{\dfrac{[(n+1)!\cdot(n+2-1)]\cdot \diagup\!\!\!\!n!}{(n+1)!\cdot (n+1)\cdot \diagup\!\!\!\!n!}}\\\\=\\\\\mathsf{\dfrac{(n+1)!\cdot(n+1)}{(n+1)!\cdot (n+1)}}\\\\=\\\\\boxed{\mathsf{1}}$


O resultado da simplificação é igual a 1!

Dúvidas? comente