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If `S_(1), S_(2), S_(3),...,S_(n)` are the sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1)` respectively, then find the values of `S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`. |
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Answer» Correct Answer - `(1)/(6) (2n) (2n +1) (4n +1) -1` Consider an infinite GP with first term 1,2,3,.., n and common ratios `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n +1)` `:. S_(1) = (1)/(1 -1//2) = 2` `{:(" "S_(2) = (2)/(1 -1//3) = 3),(" "vdots " "vdots " "vdots),(S_(2n -1) = (2n -1)/(1-1//2n) = 2n):}` `:. S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)` `= 2^(2) + 3^(2) + 4^(2) + ...+ (2n)^(2)` `= (1)/(6) (2n) (2n+1) (4n+1) -1` |
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