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) 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))

Show that sum S_(n) of n terms of an AP with first term a and common difference d is S_(n)=(n)/(2)(2a+(n-1)d)

If S_(1),S_(2),......S_(p) are the sum of n terms of an A.P.whose first terms are 1,2,3,...... and common differences are 1,3,5,7..... Show that S_(1)+S_(2)+......+S_(p)=(np)/(2)[np+1]

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_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),...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_(n) denotes the sum to terms of an A.P. whose first term is a . If the commom difference d is given by d = S_(n) - kS_(n-1) + S_(n-2) , then k is equal to

If S_(1),S_(2),...,S_(n) are the sum of n term of n G.P.whose first term is 1 in each and common ratios are 1,2,3,...,n respectively, then prove that S_(1)+S_(2)+2S_(3)+3S_(4)+...(n-1)S_(n)=1^(n)+2^(n)+3^(n)+...+n^(n)