Since, 7 th term of HP is $\frac{1}{10}$ and 12 th term is $\frac{1}{25}$.
Its corresponding terms of $AP$ is 10 and 25 .
$\therefore \, T_{7}=a+(n-1) d=10$
$\Rightarrow \, a+(7-1) d=10 $
$ \Rightarrow \, a+6 d=10\,\,\,\,\,\,\dots(i)$
and $T_{12}=a+(12-1) d=25$
$\Rightarrow \, a+11 d=25 \,\,\,\,\,\dots(ii)$
On solving Eqs. (i) and (ii), we get
$ a =-8, \,d=3 $
$\therefore \, T_{20} =-8+(20-1) 3 $
$=-8+57$
$=49$
Then, 20 th term of $HP$ is $\frac{1}{49}$.