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_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=(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=(m n)/2(m n+1)

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) .

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) .

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) .

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_(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, 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)