The $nth$ term of the sequence $2\frac{1}{2},1\frac{7}{13},1\frac{1}{9},\frac{20}{23}.$ is 

The given sequence can be written as  

$\frac{5}{2},\frac{20}{13},\frac{10}{9},\frac{20}{23},\cdots$

The reciprocal of the number

$\frac{8}{20},\frac{13}{20},\frac{18}{20},\frac{23}{20},\cdots$

This is an A.P

$nth$ term of this A.P. is

$a_n=\frac{8+5(n-1)}{20}=\frac{3+5n}{20}$

Thus, $nth$ term of the given sequence is 

$\frac{20}{5n+3}$