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_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_(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"}=

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 sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are (1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1) respectively, then find the values of S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2) .

If S _(1), S _(2) , S _(3)……., S _(2n) are the sums of infinite geometric series whose first terms are respectively 1,2,3,…..,2n and common ratio are respectively, 1/2, 1/3, …….., (1)/(2n +1), find the value of , S_(1) ^(2) + S_(2) ^(2) +…....+ S _(2n -1) ^(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_3s_1-s_1s_2-s_2s_3)/(s_1-2s_2+s_3)

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

If S_1, S_2 ,S_3,.........S_n,........ are the sums of infinite geometric series whose first terms are 1,2,3............n,............. and whose common ratio 1/2,1/3,1/4,........,1/(n+1),.... respectively, then find the value of sum_(r=1)^(2n-1) S_1^2 .

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