Question

Simpiflique as expressões

48!+49!
_______
50!


n!
___
(n+1)!


(n+3)! .(n-1)!
____________
(n-2)! (n+2)!

Answer 1

$\boxed{\frac{48!+49!}{50!}=\frac{48!+49.48!}{50*49*48!}=\frac{48!(1+49)}{50*49*48!}=\frac{1}{49}}$

$\boxed{\frac{n!}{(n+1)!}=\frac{n!}{(n+1)n!}=\frac{1}{n+1}}$

$\frac{(n+3)!(n-1)!}{(n-2)!(n+2)!}=\frac{(n+3)(n+2)(n+1)n(n-1)(n-2)!(n-1)!}{(n-2)!(n+2)(n+1)n(n-1)!}=\\ \\ =(n+3)(n-1)$

Answer 2

a) $\dfrac{48!+49!}{50!}=\dfrac{48!\cdot(1+49)}{50\cdot49\cdot48!}=\dfrac{50}{50\cdot49}=\dfrac{1}{49}$


b) $\dfrac{n!}{(n+1)!}=\dfrac{n!}{(n+1)n!}=\dfrac{1}{n+1}$

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