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_(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)=(mn)/(2)(mn+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), …, S_(m) are the sums of n terms of m A.P.'s whose first terms are 1, 2, 4, …, m and whose common differences are 1, 3, 5, …, (2m-1) repectively, then show that S_(1)+S_(2)+S_(3)+…+S_(n)=(1)/(2)mn(mn+1)

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_(p) are the sums of n terms of 'p' A.P's whose first terms are 1,2,3…p and common differences are 1,3,5… (2p-1) respectively then s_(1) + s_(2) + s_(3) + …+ s_(p) =

If S_1 , S_2 ,....... S_p are the sum of n terms of an A.P.,whose first terms are 1,2,3,...... and common differences are 1,3,5,7.....Show that S_1 + S_2 +.......+ S_p = [np]/2[np+1]

If S_(1),S_(2),......S_(p) are the sum of n terms of an A.P.whose first terms are 1,2,3,...... and common differences are 1,3,5,7..... Show that S_(1)+S_(2)+......+S_(p)=(np)/(2)[np+1]

If S_(1),S_(2),S_(3),...,S_(m) are the sums of the first n terms of m arithmetic progressions,whose first terms are 1,4,9,16,...,m^(2) and common difference are 1,2,3,4,...,m respectively,then the value of S_(1)+S_(2)+S_(3)+...+S_(m):

If S_1,S_2,S_3,...,S_n be the sums of first n terms of n G.P.'s whose first terms are each unity and the common ratios are 1, 2, 3,.....,n respectively, prove that S_1+S_2+2S_3+3S_4+.........+(n-1)S_n=1^n+2^n+3^n+....+n^n.

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) .