If pth,qth and rth term of a G.P. are x,y and z respectively prove that :
xa−rxyr−pxzp−q=1.
Given,
arp−1=x
arq−1=y
arr−1=z
⇒(arp−1)q−r=xq−r
⇒(arq−1)r−p=yr−p
⇒(arr−1)p−q=zp−q
xq−ryr−pzp−q=(arp−1)q−r(arq−1)r−p(arr−1)p−q
xq−rxyr−pxzp−q=a(p+q+r−p−q−r)r(pq+qr+rp+p+q+r−qp−rq−pr−p−q−r)
⇒xq−rxyr−pxzp−q=a0r0
∴xq−rxyr−pxzp−q=1
Hence proved.