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^ndot`

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=(m n)/2(m n+1)

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=(m n)/2(m n+1)

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)

The sum of n, 2n, 3n terms of an A.P. are S_1, S_2, S_3 respectively. Prove that S_3 = 3(S_2 - S_1)

If S_(n) denote the sum to n terms of an A.P. whose first term is a and common differnece is d , then S_(n) - 2S_(n-1) + S_(n-2) is equal to

If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

The sum of n ,2n ,3n terms of an A.P. are S_1S_2, S_3, respectively. Prove that S_3=3(S_2-S_1)dot

If the sum of n, 2n, 3n terms of an A.P are S_(1), S_(2), S_(3) , respectively, prove that S_(3) = 3 (S_(2) -S_(1)).

If the sum of n, 2n and infinite terms of G.P. are S_(1),S_(2) and S respectively, then prove that S_(1)(S_(1)-S)=S(S_(1)-S_(2)).

Let S_n denote the sum of n terms of an AP whose first term is a. If common difference d is given by d=Sn-kS_(n-1)+S_(n-2) , then k is :