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)`

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_1 , s_2 , s_3 are thesums of n natural numbers, theirsquares, their cubes respectively. Show.that 9s_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 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)) .

If s_1 ,s_2 and s_3 are the sum of first n,2n,3n terms respectively of an arithmetic progression, then show that s_3=3(s_2-s_1) .

The parabolas y^2=4xa n dx^2=4y divide the square region bounded by the lines x=4,y=4 and the coordinate axes. If S_1,S_2,S_3 are the areas of these parts numbered from top to bottom, respectively, then (a) S_1: S_2-=1:1 (b) S_2: S_3-=1:2 S_1: S_3-=1:1 (d) S_1:(S_1+S_2)=1:2

The parabolas y^2=4xa n dx^2=4y divide the square region bounded by the lines x=4,y=4 and the coordinate axes. If S_1,S_2,S_3 are the areas of these parts numbered from top to bottom, respectively, then S_1: S_2-=1:1 (b) S_2: S_3-=1:2 S_1: S_3-=1:1 (d) S_1:(S_1+S_2)=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_(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)

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) .