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_3=2S_2`.

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))

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 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 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)

If S_1, S_2, S_3,…..S_p are the sum of infinite geometric series whose first terms are 1,2,3,…p and whose common ratios are 1/2,1/3,1/4,….1/(p+1) respectively, prove that S_1+S_2+……..+S_p=p(p+3)/2

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

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)=(mn)/(2)(mn+1)

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