For some αϵR, if α−xpx=α−yqy=α−zrz and p,q,r are in A.P., then
x, y, z are in H.P.
x, y, z are in G.P.
x2,y2,z2 are in A.P.
No such relation in x, y, z
Let α−xpx=α−yqy=α−zrz=λ
⇒p=α−xλx,q=α−yλy,r=α−zλz
Given, 2q=p+r⇒2αλy−2λ=αλx−1λ+αλz−1λ
⇒2y=1x+1z