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 HP, prove that `n=(2S_(3)S_(1)-S_(1)S_(2)-S_(2)S_(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_3s_1-s_1s_2-s_2s_3)/(s_1-2s_2+s_3)

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_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)

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 :

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 :

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 :

If S_(n)=n^(2)a+(n)/(4)(n-1)d is the sum of the first n terms of an arithmetic progression, then the common difference is

Let S_(n),n=1,2,3,"…" be the sum of infinite geometric series, whose first term is n and the common ratio is (1)/(n+1) . Evaluate lim_(n to oo)(S_(1)S_(n)+S_(2)S_(n-1)+S_(3)S_(n-2)+"..."+S_(n)S_(1))/(S_(1)^(2)+S_(2)^(2)+"......"+S_(n)^(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