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)`.

Let S_(1),S_(2) and S_(3) be the sum of n terms of 3 arithmetic series,the first termof each being 1 and the respective common differences are 1,2,3, then prove that S_(1)+S_(2)+2S_(2)

If S_(1),S_(2),......,S_(p) are the sums of n terms of an A.P

If {S_1,S_2,S_3} denoted the sum of n terms of three series in A.P. the first term of each being the same and the respective common difference 1, 2 and 3, show that {S_1+S_3=2S_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, 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_3=K,S_2 , where S_1,S_2 and S_3 are the sum of 'n' terms of the series in A.P .The first term of each being one and the respective common difference being 1,2,3 then find k=………..

The sum of first n terms of three APs are S_1, S_2 and S_3 . The first term of each AP is 5 and their common differences are 2, 4 and 6 respectively. Prove 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) be the sum of 10, 20, 30 terms respectively of an A.P. Then