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

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) and S_(3) are the sums of first n natureal numbers, their squares and their cubes respectively, then (S_(1)^(4)S_(2)^(2)-S_(2)^(2)S_(3)^(2))/(S_(1)^(2) +S_(2)^(2))=

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 sums of n terms of three arithmetical progressions are S_1, S_2 and S_3dot The first term of each unityand the common differences are 1,2 and 3 respectively. Prove that S_1+S_3=2S_2dot

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

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

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

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

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. 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) equals :