The sums of `n` terms of three arithmetical progressions are `S_1, S_2a n dS_3dot` The first term of each unityand the common differences are `1,2a n d3` 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_(3). The first term of each unity and the common differences are 1,2 and 3 respectively.Prove that S_(1)+S_(3)=2S_(2)

If the sum of 'n' terms of an arithmetic progression is S_(n)=3n +2n^(2) then its common difference is :

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),...,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):

Let S_(1),S_(2),S_(3) be the respective sums of first n,2n and 3n terms of the same arithmetic progression with a as the first term and d as the common difference.If R=S_(3)-S_(2)-S_(1) then R depends on

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 the sum of n,2n,3n terms of an AP are S_(1),S_(2),S_(3) respectively.Prove that S_(3)=3(S_(2)-S_(1))

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_(3)s_(1)-s_(1)s_(2)-s_(2)s_(3))/(s_(1)-2s_(2)+s_(3))

If the sum of the first n terms of an AP is given by S_n=3n^2-n then find its nth terms, first term and common difference