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)`

S_(n) is the sum of n terms of an A.P. If S_(2n)= 3S_(n) then prove that (S_(3n))/(S_(n))= 6

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 the sum of n terms of an A.P. is 3n + 2n^(2) , find the common difference

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

The ratio of the sums of m and n terms of an A.P. is m^2 : n^2. Show that the ratio of m^(th) and n^(th) term is (2m-1) : (2n -1) .

The sum of first n terms of three APs are S_1, S_2 and S_3 . The first term of each AP is 5 and their common differences are 2, 4 and 6 respectively. Prove that S_1 + S_3 = 2S_2.

S_(n) denots the sum of first n terms of an A.P. Its first term is a and common difference is d. If d= S_(n)-k S_(n-1) + S_(n-2) then k= ……….

The mean and standard deviation of a set of n_1 observation are barx_1 and S_1 , respectively while the mean standard deviation of another set of n_2 observations are barx_2 and S_2 , respectively . Show that the standard deviation of the combined set of (n_1+n_2) observations is given by standard deviation sqrt ((n_1(S_1)^2 + n_1(S_2)^2)/(n_1+n_2) + (n_1+n_2 (barx_1 - barx_2)^2)/(n_1-n_2)^2)

C_1 is a curve y^2=4x,C_2 is curve obtained by rotating C_1,120^@ in anti -clockwise direction C_3 is reflection of C_2 with respect to y=x and S_1,S_2,S_3 are foci of C_1,C_2 and C_3 , respectively, where O is origin. IF S_1(x_1,y_1),S_2(x_2,y_2) and S_3(x_3,y_3) then the value of sumx_1^2+sumy_1^2 is

C_1 is a curve y^2=4x,C_2 is curve obtained by rotating C_1,120^@ in anti -clockwise direction C_3 is reflection of C_2 with respect to y=x and S_1,S_2,S_3 are foci of C_1,C_2 and C_3 , respectively, where O is origin. Area of DeltaOS_2S_3 is