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_(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_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.

Let sum of n , 2n , 3n , terms of an A.P are S_(1), S_(2), S_(3) respectively. Prove that S_(3) = 3 (S_(2) - S_(1)) .

The sum of first n, 2n and 3n terms of an A.P. are S_(1), S_(2), S_(3) respectively. Prove that S_(3)=3(S_(2)-S_(1)) .

If S_(n) denotes the sum of n terms of an AP whose common difference is d, the value of S_(n)-2S_(n-1)+S_(n-2) is

If S_(n) denotes the sum of n terms of an A.P., S_(n +3) - 3S_(n + 2) + 3S_(n + 1) - S_(n) =

Let S_(n) denote the sum of first n terms of an A.P. If S_(4) = -34 , S_(3) = - 60 and S_(6) =- 93 , then the common difference and the first term of the A.P. are respectively.