If the pth,qth terms of a G.P. are q,p respectively ,show that (p+q)th term is (qp/pq)1/p-q
Ap = arp-1 = q ---->(1)
Aq = ar q-1 =p ----->(2)
dividing (1) & (2)
r p-1 / rq-1 = q/p
rp- q = q/p ---> r = (q/p)1/ p-q
then arp-1 = q
a ((q/p)1/ p-q)p-1 = q
a((q/p)p-1/ p-q) = q
a = q / (q/p)p-1/ p-q)
now Ap+q = arp+q -1
= (q / (q/p)p-1/ p-q) .((q/p)1/ p-q)p+q -1
= (q 1 - (p-1)/ p-q . p p-1 / p-q ) ( q p+q-1 / p-q / p p+q-1 / p-q )
= ( q p-q - p +1 /p-q pp-1/ p-q ) ( q p+q-1 / p-q / p p+q-1 / p-q )
= q1-q / p-q + (p+q-1)/p-q . p p-1 / p-q - (p+q-1/ p-q)
= qp/ p-q p q/ p-q
= (qp / pq) 1/ p-q