a, b, c, d are in G.P ⇒ b = ar, c = ar2, d = ar3, where r = common ratio
∴ (ab + bc + cd)2 = (a.ar + ar . ar2 + ar2 . ar3)2 = [a2r (1 + r2 + r4)]2 ...(i)
(a2 + b2 + c2) (b2 + c2 + d2) = (a2 + a2r2 + a2r4) (a2r2 + a2r4 + a2r6)
= a2 (1 + r2 + r4) . a2r2 (1 + r2 + r4) = a4r2 (1 + r2 + r4)2
= [a2r (1 + r2 + r4)]2 = (ab + bc + cd)2 (From (i))
∴ (a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.