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_3..........,S_q are the sums of n terms of q ,AP's whose first terms are 1,2,3,...........q and common difference are 1,3,5,...........,(2q-1) respectively ,show that S_1 + S_2 + S_3..........+S_q=1/2nq(nq+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_(n) are the sum of n term of n G.P.whose first term is 1 in each and common ratios are 1,2,3,...,n respectively, then 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),….., 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"}=

Show that sum S_(n) of n terms of an AP with first term a and common difference d is S_(n)=(n)/(2)(2a+(n-1)d)