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Question

If S1,S2,S3 denote the sum of first 11 terms of three arithmetic progressions whose first terms are unity and their common differences are in harmonic progressions, then the value of (2S3S1S1S2S2S3S12S2+S3) is

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Solution

Let their common differences be d1,d2,d3
Given d1,d2,d3 are in H.P.
1d1,1d2,1d3 are in A.P.
S1=112[2+(111)d1]=11(1+5d1)
d1=S11155

Similarly, d2=S21155,d3=S31155
55S111,55S211,55S311 are in A.P.
55(1S2111S111)=55(1S3111S211)
S1S2S111=S2S3S311
11(S12S2+S3)=2S3S1S1S2S2S3
2S3S1S1S2S2S3S12S2+S3=11

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