#1/(n!)+1/((n+1)!)+1/((n+2)!)=# ?

1 Answer
George C.
Jul 28, 2018

#1/(n!)+1/((n+1)!)+1/((n+2)!)=(n^2+4n+5)/((n+2)!)#

Explanation:

By the definition of factorials we have:

#(n+1)! = n!(n+1)#

and:

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

So:

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

#=((n+1)(n+2))/((n+2)!)+(n+2)/((n+2)!)+1/((n+2)!)#

#=((n+1)(n+2)+(n+2)+1)/((n+2)!)#

#=(n^2+3n+2+n+2+1)/((n+2)!)#

#=(n^2+4n+5)/((n+2)!)#

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