If a, b,and c are in A.P., p, q, and r are in H.P. and ap, bq, and cr are in G.P., then pr+rp is equal to
pr+rp=p2+r2pr=(p+r)2−2prpr
=4p2r2q2−2prpr
=4prq2−2=4b2ac−2
[∵ap,bq,cr are in A.P.⇒b2q2=acpr]
=(a+c)2ac−2[a,b,c are in A.P⇒.2b=a+c]
ac+ca