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)`.

If S_n denotes the sum of first n terms of an AP, prove that S_(12) = 3 (S_(8) -S_4)

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

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)

A monochromatic light source of wavelength lamda is placed at S. three slits S_(1),S_(2) and S_(3) are equidistant from the source S and the point P on the screen S_(1)P-S_(2)P=lamda//6 and S_(1)P-S_(3)P=2lamda//3 . if I be the intensity at P when only one slit is open the intensity at P when all the three slits are open is-

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.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^(2)R^(n) = S^n .

For an A.P., S_(1)=6 and S_(7)= 105 . Prove that S_(n): S_(n-3) = (n+ 3): (n-3)

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 in an AP, S_(n)= q n^(2) and S_(m)= qm^(2) , where S_(r ) denotes the sum of r terms of the AP, then S_(q) equals to,

For as Ap. S_(n)= m and S_(m) = n . Prove that S_(m+n)=-(m+n).(m ne n)