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Question

If S1,S2,....Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ...., n respectively, then prove that. S1+S2+2S3+3S4+....(n−1) Sn=1n+2n+3n+...+nn.

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Solution

Given, S1,S2,.....,Sn are the same of n terms of an G.P. whose first term is 1 in each case and the common ratios are 1, 2, 3, ...., n. ∴S1=1+1+1+..... n terms = n .....(i) S2=1(2n−1)2−1=2n−1 ...........(ii) S3=1(3n−1)3−1=3n−12 ........ (iii) S4=1(4n−1)4−1=4n−13 ....... (iv) Sn=1(nn−1)n−1=nn−1n−1 ......(n) Now, LHS = S1+S2+2S3+3S4+....+(n−1)Sn =n+2n−1+3n−1+4n−1+....+nn [Using (1), (2), (3), ....., (n)] =n+(2n+3n+4n+.....+nn)−[1+1+1+....+(n−1)times] =n+(2n+3n+4n+....+nn)−(n−1) =n+(2n+3n+4n+....+nn)−n+1 =1+2n+3n+4n+....+nn =1n+2n+3n+4n+....+nn = RHS Hence proved.

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