If `s_1,s_2,s_3` denote the sum of n terms of 3 arithmetic series whose first terms are unity and their common difference are in H.P., Prove that `n=(2s_3s_1-s_1s_2-s_2s_3)/(s_1-2s_2+s_3)`

IF S_1,S_2,S_3 denote the sum n(gt1) terms of three sequences in A.P., whose first terms are unity and common differences are in H.P.prove that n=(2S_3S_1-S_1S_2-S_2S_3)/(S_1-2S_2+S_3)

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_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_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 and S_3 are respectively the sums of n, 2n and 3n terms of an A.P, whose first term is a and common difference d, then find S_1,S_2 and S_3

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) =

The sums of n terms of three arithmetical progressions are S_1, S_2 and S_3dot The first term of each unity and the common differences are 1,2 and 3 respectively. Prove that S_1+S_3=2S_2dot

The sums of n terms of three arithmetical progressions are S_1, S_2 and S_3dot The first term of each unity and the common differences are 1,2 and 3 respectively. Prove that S_1+S_3=2S_2dot

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.

The sums of n terms of three arithmetical progressions are S_1, S_2 and S_3dot The first term of each unityand the common differences are 1,2 and 3 respectively. Prove that S_1+S_3=2S_2dot