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

Question

5644 views around the world

You can reuse this answer
Creative Commons License

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 $(n+1)/(n^2+3)$