If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that S12 + S22 = S1(S2 + S3)

Question May be wrong.

Sum of GP for n terms S1 =


= …(1)


Sum of GP for 2n terms S2 =


= …(2)


Sum of GP for 3n terms S3 = …(3)


=


Let


∴ 1, 2 and 3 becomes


S1 = K(rn - 1)


S2 = K(r2n - 1)


S3 = K(r3n - 1)


∴ S12 + S22 = k2(rn - 1)2 + k2(r2n - 1)2


= k2(r2n + 1 - 2.rn + r4n + 1 - 2.r2n)


= k2(r4n - r2n - 2rn + 2)


L.H.S = k2(r4n - r2n - 2rn + 2)


S1(S2 + S3) = K(rn - 1)[ (K(r2n - 1) + K(r3n - 1))]


= K2(rn - 1)[r2n + r3n - 2]


= k2(r4n - r2n - 2rn + 2)


Hence, Proved.


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