$\text{ If the }{p}^{\text{th }},q^{th}\text{, and }{r}^{\text{th }}\text{ terms of an A.P. are in G.P. then common ratio of the G.P is}$

$p^{\text{th }},q^{\text{th }},r^{\text{th }}\text{ terms of A.P are }$

$\begin{array}{l}a+(p-1)d=x\\ a+(q-1)d=xR\\ a+(r-1)d=x{R}^2\\ \text{ Where }\mathrm{R}\text{ is common ratio of G.P }\end{array}$

Subtracting   (3) from (2) and (2) from (1) and then dividing the former
by the later, we have $\frac{q-r}{p-q}=\frac{x{R}^2-xR}{xR-x{R}^2}=R$