Let first four and common ratio of G.P series be a and r respectively
∴pthterm=q⇒arp−1=q........1
qthterm=p⇒arq−1=p..........2
1÷2
arp−1arq−1=pq
or, arp−1−q+1=pq⇒r=(qp)1p−q
Putting value of r in 1
arp−1=q
Or, a(qp)p−1p−q=q
Or, a=q(pq)p−1p−q
∴p+qthterm=arq−1
=q(pq)p−1p−q(qp)p+q−1p−q
=q⎛⎝1−p−1p−q+p+q+−1p−q⎞⎠×p⎛⎝p−1p−q−p+q+−1p−q⎞⎠
=q⎛⎝p−q−p+1+p+q−1p−q⎞⎠×p⎛⎝p−1−p−q+1p−q⎞⎠
qpp−q×q−qp−q
=(qppq)1p−q Ans