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If a, b, c, d are in G.P., prove that (a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.

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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.

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