If `S_(1),S_(2),S_(3),...S_(m)` be the sum of first n terms of n A.P. Whose first terms are `1,2,3,... M` aand common differnece are `1,3,5,...(2m-1)` respectively, show that `S_(1)+S_(2)+S_(3)+...+S_(m)= (mn)/(2)(mn+1)`.

If S_1,S_2, S_3, S_m are the sums of n terms of m A.P. ' s whose first terms are 1,2,3, ,m and common differences are 1,3,5, ,(2m-1) respectively. Show that S_1+S_2,+S_m=(m n)/2(m n+1)

If S_1,S_2...S_m denote the sums of n terms of m numbers of A.P is whose first terms are 1,2,…m and common differences are 1,3,..(2m-1)respectively,Show that S_1+S_2+...+S_m=(mn)/2(mn+1)

If S_(1), S_(2), S_(3) be the sums of n terms of three aA.P.'s, the first term of each A.P. being 1 and the respective common differences are 1, 2, 3 then show that, S_(1) + S_(3) = 2S_(2) .

The sum of n term of three A.P are S_1,S_2, and S_3 . The frist term of each is 1 and common diffrence are 1,2,3 repectively .Prove that S_1+S_3 =2 S_2.

If S_(1), S_(2), S_(3),…., S_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

If S_(1), S_(2), S_(3),...S_(n) are the sums of infinite geometric series, whose first terms are 1,2,3,...n whose ratios are (1)/(2),(1)/(3),(1)/(4) ,...(1)/(n+1) respectively, then find the value of S_(1)^(2)+S_(2)^(2)+S_(3)^(2)+...+S_(2n-1)^(2) .

If S_n=n P+(n(n-1))/2Q ,w h e r eS_n denotes the sum of the first n terms of an A.P., then find the common difference.

If the sums of n, 2n and 3n terms of an A.P. be S_(1), S_(2), S_(3) respectively, then show that, S_(3) = 3(S_(2) - S_(1)) .

If S_(n) =nP +1/2 n (n-1) Q, where S_(n) is the sum of first n terms of an A.P. then the common difference of the A.P. will be-

Given p no. of A.P. each of which consists of an n terms. If their first terms are 1,2,3.... p are in common differences are 1,3,5...2p-1 respectively, then the sum of the terms of all progressions is