Q57 of 74 Page 9

If a, b, c, d are in G.P., show that (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)2


Since a, b, c, d are in G.P

b/a = c/b = d/c = r, where r is the common ratio.

∴ b = ar, c = br = ar2 and d = cr = ar3.

L. H. S = (a2 + b2 + c2)(b2 + c2 + d2)

          = [a2 + (ar)2 + (ar2)2][( ar)2 + (ar2)2 + (ar3)2]

          = (a2 + a2r2 + a2r4 )(a2r2 + a2r4 + a2r6)

          = a4 r2 (1 + r2 + r4 )(1 + r2 + r4)

          = a4 r2 (1 + r2 + r4 )2

R.H.S = (ab + bc + cd)2

          = (a ar + ar  ar2 + ar2 ar3)2

        = (a2r + a2r3 + a2r5)2

        = (a2r)2(1 + r2 + r3)2

        = a4 r2 (1 + r2 + r4 )2

L.H.S = R. H. S

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