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 be the sums of first n term ,of 2 n terms and of 3n terms of a G.P. respectively,then prove that- S_1^2+S_2^2=S_1(S_2+S_3) .

If S_1,S_2 and S_3 be the sums of first n term ,of 2 n terms and of 3n terms of a G.P. respectively,then prove that- S_1(S_3-S_2)=(S_2-S_1)^2 .

If the sum of first n,2n,3n terms of an A.P. be S_1 , S_2 and S_3 respectively then prove that S_3=3(S_2-S_1)

If the sum of first n,2n,3n terms of an A.P. be S_1 , S_2 and S_3 respectively then prove that S_1+S_3=2S_2

In a G.P the sum of first n terms is S,the product is P and the sum of the reciprocals of the terms is R.Show that P^2=(S/R)^n .

If {S_1,S_2,S_3} denoted the sum of n terms of three series in A.P. the first term of each being the same and the respective common difference 1, 2 and 3, show that {S_1+S_3=2S_2} .

If S_n denotes the sum of the first n terms of an AP and if S_2n=3S_n then S_3n:S_n is equal to

If S_n denotes the sum of first n terms of an A.P. prove that S_12=3(S_8-S_4) .

The sum of first n terms of an A.P. is S_n .If S_n=n^2p and S_m=m^2p(mnen) .Show that S_p=p^3 .

If A be the A.M. of two given numbers and g_1 and g_2 be two G.M.'s between them,then show that- 2A=(g_1^2)/(g_2)+(g_2^2)/g_1 .