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If s1,s2,s3,………s(2n) are the sums of inf...

If `s_1,s_2,s_3,………s_(2n)` are the sums of infinite geometric series whose first terms are respectively 1,2,3,…2n and common ratioi are respectively `1/2, 1/3, …………, 1/(2n+1)` find the value of `s_1^2+s_2^2+……….+s_(2n-1)^2`

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If S _(1), S _(2) , S _(3)……., S _(2n) are the sums of infinite geometric series whose first terms are respectively 1,2,3,…..,2n and common ratio are respectively, 1/2, 1/3, …….., (1)/(2n +1), find the value of , S_(1) ^(2) + S_(2) ^(2) +…....+ S _(2n -1) ^(2).

If S_(1),S_(2),S_(3),"…….",S_(2n) are the same of infinite geometric series whose first terms are respectively 1 , 2, 3,"…….", 2n and common ratio are respectively. 1/2, 1/3, '…..". (1)/(2n+1) .Find the value of S_(1)^(2) + S_(2)^(2) + "……" + S_(2n-1)^(2) .

Knowledge Check

  • If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

    A
    `1/15`
    B
    `1/60`
    C
    `1/12`
    D
    `1/3`

  • If S_(1), S_(2),cdots S_(n) , ., are the sums of infinite geometric series whose first terms are 1, 2, 3,…… ,n and common ratios are (1)/(2) ,(1)/(3),(1)/(4),cdots,(1)/(n+1) then S_(1)+S_(2)+S_(3)+cdots+S_(n) =

    A
    `(n(n+1))/(2)`
    B
    `((n+1)(n+3))/(2)`
    C
    `(n(n+2))/(2)`
    D
    `(n(n+3))/(2)`

  • If S_(1), S_(2),…S_(lambda) are the sums of infinite G.P.'s whose first terms are respectively 1, 2, 3,…. lambda and common ratios are (1)/(2), (1)/(3),…(1)/(lambda + 1) respectively, then S_(1) + S_(2) + S_(3) +...+ S_(lambda) =

    A
    `(lambda (lambda + 1))/(2)`
    B
    `(lambda (lambda + 2))/(2)`
    C
    `(lambda (lambda + 3))/(2)`
    D
    none

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