Thank you for reporting, we will resolve it shortly
Q.
If in a progression $a_1,a_2,a_3, ... $ etc. $(a_r=a_{r+1})$ bears a constant ratio with $a_r.a_{r+1},$ then the terms of the progression are in
Sequences and Series
Solution:
Since $\frac{a_{r}-a_{r+1}}{a_{r}. a_{r+1}} $ constant for all $r $
$ \Rightarrow \frac{1}{a_{r+1}}-\frac{1}{a_{r}} = $ constant for all $ r$
$ \Rightarrow a_{1}, a_{2}, a_{3}, .....$ are in $H.P$.