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)`.

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

Find the sum of first n terms of an A.P. whose nth term is 3n+1.

If the coefficient of the rth, (r+1)th and (r+2)th terms in the expansion of (1+x)^(n) are in A.P., prove that n^(2) - n(4r +1) + 4r^(2) - 2=0 .

If the first and the nth term of a G.P. are a and b. respectively, and if P is the product of n terms, prove that P^2 = (ab)^n .

Find the sum S_n of the cubes of the first n terms of an A.P. and show that the sum of first n terms of the A.P. is a factor of S_n .

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n+ 1)th to (2n)th term is 1/r^n .

If the first and the n^(t h) term of a GP are a and b, respectively, and if P is the product of n terms, prove that P^2=(a b)^n .

Prove that sum_(r=0)^n 3^r "^nC_r= 4^n .

In the expansion of (x+a)^n if the sum of odd terms is P and the sum of even terms is Q , then a. P^2-Q^2=(x^2-a^2)^n b. 4P Q=(x+a)^(2n)-(x-a)^(2n) c. 2(P^2+Q^2)=(x+a)^(2n)+(x-a)^(2n) d. none of these

If S_1,S_2,S_3 be the sum of n, 2n, 3n terms of a G.P., show that : S_1(S_3-S_2)= (S_2-S_1)^2 .