If the pth,qth and rth terms of a G.P. are a,b and c, respectively. Prove that aq−rbr−pcp−q=1.
Let a be the first term and r be the common ratio of the G.P.
Thus according to the given information,
arp−1=a,arp−1=b,arp−1=c
∴aq−rbr−pcp−q
=aq−r×r(p−1)(q−r)×ar−p×r(q−1)(r−p)×ap−q×r(r−1)(p−q)
=a−r+r−p+p−q×r(pr−pr−q+r)+(rq−r+p−pq)+(pr−p−qr+q)
=a0⋅r0=1. [henceproved]