If `S_(1), S_(2), S_(3)` are the sum of first n natural numbers, their squares and their cubes respectively. Show that `9S_(2)^(2) =S_(3)(1+8S_(1))`

If S_1 , S_2 , S_3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that 9 S_2^2=S_3(1+8S_1) .

Show that S_(n)=(n(2n^(2)+9n+13))/(24) .

Let the sum of n, 2n, 3n terms of an A.P. be S_1 , S_2 and S_3 , respectively, show that S_3 = 3(S_2 - S_1)

If S_n represents the sum of n terms of a G.P. whose first term and common ratio are a and r respectively. Prove that : S_1 +S_2+S_3+.....+S_(m)= (am)/(1-r)-(ar(1-r^(m)))/((1-r)^2) .

Calculate the oxidation number of S and C respectively in, (i) S_(2)O_(7)^(-2) (ii) CO_(2)

If for a sequence {a_(n)},S_(n)=2n^(2)+9n , where S_(n) is the sum of n terms, the value of a_(20) is

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 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) be respectively the sum of n, 2n and 3n terms of a GP, then (S_(1)(S_(3)-S_(2)))/((S_(2)-S_(1))^(2)) is equal to

If S_n denotes the sum of n terms of a G.P. whose first term and common ratio are a and r respectively, then : S_1 +S_3+S_5+.....+S_(2n-1)= (an)/(1-r)-(ar(1-r^(2n)))/((1-r)^2 (1+r)) .