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)`

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 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 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) .

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

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 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 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), S_(3),...S_(n) are the sums of infinite geometric series, whose first terms are 1,2,3,...n whose ratios are (1)/(2),(1)/(3),(1)/(4) ,...(1)/(n+1) respectively, then find the value of S_(1)^(2)+S_(2)^(2)+S_(3)^(2)+...+S_(2n-1)^(2) .