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Question

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).

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Solution

Let a be the first term and r be the common ratio of the given G.P.

Sum of n terms, S1=arn-1r-1 ... 1Sum of 2n terms, S2 =ar2n-1r-1S2 =arn2-12r-1S2 =arn-1rn+1r-1S2 =S1rn+1 ....2And, sum of 3n terms, S3= ar3n-1r-1S3=arn3-13r-1S3=arn-1r2n+rn+1r-1S3=S1r2n+rn+1 ...3

Now, LHS=S12 + S22 =S12 + S1rn+12 Using 2=S121 + rn+12=S121+r2n+2rn+1=S12r2n+rn+1+rn+1=S1S1r2n+rn+1+S1rn+1=S1S2+S3 Using 2 and 3=RHSHence proved.

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