Ques) Let a₁,a₂,a₃,..... be in A.P and a_p, a_q, a_r be in G.P. Then show that ap / aq = (q-p) / (r - q).

since, a_p,a_q & a_r are d terms taken from the given A.P.

so,   a_{p = aq - (q-p)*d ..............................(1)}

         _{aq = ar - (r-q)*d..........................(2)}

      _{now since} a_p,a_q & a_r are in G.P. so a_q = R*a_p

                                                 _and, a_{r = R*}a_{q, where R is d common ratio of tge G.P.}

  we put  a_q = R*a_p in (1)  we get

                                                              a_p( R-1) =   (q-p)d  ............................................(3)

we put a_r = R* a_q in (2)   we get

                                                              a_q (R-1) = (r-p) d       .............................................(4)

                                              
dividing  (3) by (4) we get

                                    a_p/a_q = (q-p)/ (r-p)