Question

simplifique as expressões. a) n!/(n-1)!
b) (n+2)!/(n+3)!
c) (n+4)!/(n+2)!+(n+3)!
d) (n-1)!+n!/(n+1)!

Answer 1

Olá Jacke

a) n!/(n-1)! = n*(n-1)!/(n-1)! = n

b) (n+2)!/(n+3)! = (n+2)!/((n+3)*(n+2)!) = 1/(n+3) 

c) (n+4)!/(n+2)!+(n+3)! = n+3

d)  ((n-1)!+n!)/(n+1)! = 1/n 

.

Answer 2

a) n

b) 1/(n + 3)

c) (n + 3)

d) 1/n

a)  n!   = n.(n - 1)! = n

  (n - 1)!    (n - 1)!

b) (n + 2)! =     (n + 2)!     =    1    

    (n + 3)!    (n + 3).(n + 2)!   (n + 3)

c)        (n + 4)!       = (n + 4)!.(n + 3).(n + 2)!  = (n + 4).(n + 3).(n + 2)!

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

Eliminamos o fator (n + 2)! e sobra:

(n + 4).(n + 3) = (n + 4).(n + 3)

  (1 + n + 3)            (n + 4)

Eliminamos o fator (n + 4), fica:

(n + 3)

d) (n - 1)! + n! = (n - 1)! + n.(n - 1)! =

       (n + 1)!             (n + 1)!

Colocamos o fator (n - 1)! em evidência. Logo, fica:

(n - 1)!.(n + 1) = (n - 1)!.(n + 1) = (n - 1)! = (n - 1)! = 1

   (n + 1)!             (n + 1).n!          n!       n.(n - 1)!    n

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