If the pth ,qth, and rth  terms of an A.P. are in G.P. then common ratio of the G.P is

pth ,qth ,rth  terms of A.P are a+(p−1)d=xa+(q−1)d=xRa+(r−1)d=xR2 Where R is common ratio of G.P Subtracting   (3) from (2) and (2) from (1) and then dividing the formerby the later, we have q−rp−q=xR2−xRxR−xR2=R