Simplify #(n!)/((n+1)!)-((n-1)!)/(n!)#?

1 Answer
Evan · Ratnaker Mehta
Nov 30, 2017

#1/(n^2-n)#

Explanation:

What is #n!# ( #n# factorial ) ?

#n! =1xx2xx3xx...xxn#

So,

#(n!)/((n+1)!)-((n-1)!)/(n!)#
#=cancel(1xx2xx...xxn)/(cancel(1xx2xx...xxn)xx(n+1))-cancel(1xx2xx...xx(n-1))/(cancel(1xx2xx...xx(n-1))xxn)#
#=1/(n+1)-1/n#
#=(n-(n+1))/((n+1)(n))#
#=-1/(n^2+n)#

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