How do you find the explicit formula for the following sequence 1/2, 3/7, 1/3, 5/19, 3/14?

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Answer 1 · 2016-05-08T08:22:39

#n^(th)# term of the series #{1/2,3/7,1/3,5/19,3/14...........}# is #(n+1)/(n^2+3)#

The series #{1/2,3/7,1/3,5/19,3/14...........}# can also be written as

#{2/4,3/7,4/12,5/19,6/28,.......}#

This can be divided into two series

one #{2,3,4,5,6,...}# whose #n^(th)# term is obviously #n+1#

other is #{4,7,12,19,28,..}#, in which the difference constantly increases by #2# and is #3,5,7,9,....}#.

If one recalls this is all true for square numbers too, as in the series #{1,4,9,16,25,36,....}# the difference too increases like this.

Hence #n^(th)# term of this can be written as #n^2+3#.

Hence #n^(th)# term of the series #{1/2,3/7,1/3,5/19,3/14...........}# is #(n+1)/(n^2+3)#