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)`

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_(1), S_(2) and S_(3) denote the sum of first n_(1) , n_(2) and n_(3) terms respectively of an A.P.L , then : (S_(1))/(n_(1)) . ( n _(2) - n_(3)) + ( S_(2))/( n_(2)). ( n _(3) - n_(1)) + ( S_(3))/( n_(3)) . ( n_(1) - n_(2)) is equal to :

The sums 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) )

Let S_(n) denote the sum of the cubes of the first n natural numbers and s_(n) denote the sum of the first n natural numbers. Then sum_(r=1)^(n)(S_(r))/( S_(r )) equals :

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)

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)) .

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 first n terms of an AP and 3S_n=S_(2n) What is S_(3n):S_n equal to? What is S_(3n):S_(2n) equal to?