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

May 8, 2016

${n}^{t h}$ term of the series $\left\{\frac{1}{2} , \frac{3}{7} , \frac{1}{3} , \frac{5}{19} , \frac{3}{14.} \ldots \ldots \ldots .\right\}$ is $\frac{n + 1}{{n}^{2} + 3}$

#### Explanation:

The series $\left\{\frac{1}{2} , \frac{3}{7} , \frac{1}{3} , \frac{5}{19} , \frac{3}{14.} \ldots \ldots \ldots .\right\}$ can also be written as

$\left\{\frac{2}{4} , \frac{3}{7} , \frac{4}{12} , \frac{5}{19} , \frac{6}{28} , \ldots \ldots .\right\}$

This can be divided into two series

one $\left\{2 , 3 , 4 , 5 , 6 , \ldots\right\}$ whose ${n}^{t h}$ term is obviously $n + 1$

other is $\left\{4 , 7 , 12 , 19 , 28 , . .\right\}$, 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 $\left\{1 , 4 , 9 , 16 , 25 , 36 , \ldots .\right\}$ the difference too increases like this.

Hence ${n}^{t h}$ term of this can be written as ${n}^{2} + 3$.

Hence ${n}^{t h}$ term of the series $\left\{\frac{1}{2} , \frac{3}{7} , \frac{1}{3} , \frac{5}{19} , \frac{3}{14.} \ldots \ldots \ldots .\right\}$ is $\frac{n + 1}{{n}^{2} + 3}$