If the sums of n, 2n and 3n terms of an A.P. be `S_(1), S_(2), S_(3)` respectively, then show that, `S_(3) = 3(S_(2) - S_(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)

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 S_(1), S_(2), S_(3) be respectively the sums of n, 2n and 3n terms of a G.P., prove that, S_(1)(S_(3) - S_(2)) = (S_(2) - S_(1))^(2) .

If the first term, common ratio and the sum of first n terms of a G.P. be a, r and S_(n) respectively, find the value of S_(1) + S_(2) + S_(3) +….+S_(n) .

If S_(1), S_(2), S_(3) be the sums of n terms of three aA.P.'s, the first term of each A.P. being 1 and the respective common differences are 1, 2, 3 then show that, S_(1) + S_(3) = 2S_(2) .

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 S_(n) be the sum of first n terms of an A.P. If S_(2n) = 5s_(n) , then find the value of S_(3n) : S_(2n) .

If S_(1), S_(2), S_(3),…., S_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

If S_(1), S_(2), S_(3) are the sums of n natural numbers, their squares, their cubes respectively show that 9S_(2)^(2) = S_(3)(1+8S_(1)) .

If S_(n) be the sum of first n terms of a G.P. whose common ratio is r, then show that, (r-1)(dS_(n))/(dr)=(n-1)S_(n)-nS_(n-1)