Question

Simplifique a expressão:

(n+2)! + (n+1)! / (n+2)! - (n+1)!

Answer 1

$\dfrac{(n - 2)! + (n + 1)!}{(n + 2)! - (n + 1)!}$

$\dfrac{n!.(n + 1)(n - 2) +n!. (n + 1)}{n! .(n - 1)(n + 2)- n!(n + 1)!}\\ \\ \\\dfrac{\not n!.(n + 1)(n - 2) +\not n!. (n + 1)}{\not n! .(n - 1)(n + 2)- \not n!(n + 1)}\\ \\ \\\dfrac{(n + 1)(n - 2) +(n + 1)}{(n - 1)(n + 2)- (n + 1)}\\ \\ \\\dfrac{n^2 - n - 2 +(n + 1)}{n^2 + n - 2- (n + 1)}\\ \\ \\\dfrac{n^2 +4n + 3}{n^2 + 3n +1}\\ \\ \\\dfrac{(n + 1).(n + 3}{(n + 1) .(n+ 1}\\ \\ \\\dfrac{(\not n + \not 1).(n + 3}{(\not n + \not 1) .(n+ 1}\\ \\ \\=> \dfrac{n + 3}{n + 1}$