If `a_1,a_2, a_3……….` are in H.P. and `f(k) =sum_(r=1)^n a_r-a_k, the a_1/(f(1)), a_2/(f(2)), a_3/(f(3)), …….., a_n/(f(n))` are in (A) A.P. (B) G.P (C) H.P. (D) none of these

If a_1,a_2,a_3……..a_n are in H.P. and f(k) = sum_(r=1)^na_r-a_k then (f(1))/a_1, (f(2))/a_3 ….f(n)/a_n are (A) A.P. (B) G.P. (C) H.P. (D) none of these

If a_n= int_0^pi (sin(2n-1)x)/(sinx) dx . Then the number a_1,a_2,a_3 …….. Are in (A) A.P (B) G.P (C) H.P (D) none of these

If x,a_1,a_2,a_3,…..a_n epsilon R and (x-a_1+a_2)^2+(x-a_2+a_3)^2+…….+(x-a_(n-1)+a_n)^2le0 , then a_1,a_2,a_3………a_n are in (A) AP (B) GP (C) HP (D) none of these

If a_1,a_2,……….,a_(n+1) are in A.P. prove that sum_(k=0)^n ^nC_k.a_(k+1)=2^(n-1)(a_1+a_(n+1))

If a_1,a_2,a_3 are in G.P. having common ratio r such that sum_(k=1)^n a_(2k-1)= sum_(k=1)^na_(2k+2)!=0 then number of possible value of r is (A) 1 (B) 2 (C) 3 (D) none of these

I a_n= int_0^(pi/2) (1-cos2nx)/(1-cos2x)dx then a_1,a_2,a_3,………,a_n are in (A) A.P. only (B) G.P.only (C) H.P. only (D) A.P., G.P. and H.P.

Let a_1,a_2,a_3…………., a_n be positive numbers in G.P. For each n let A_n, G_n, H_n be respectively the arithmetic mean geometric mean and harmonic mean of a_1,a_2,……..,a_n On the basis of above information answer the following question: A_k,G_k,H_k are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

If p(x),q(x) and r(x) be polynomials of degree one and a_1,a_2,a_3 be real numbers then |(p(a_1), p(a_2),p(a_3)),(q(a_1), q(a_2),q(a_3)),(r(a_1), r(a_2),r(a_3))|= (A) 0 (B) 1 (C) -1 (D) none of these

If a_1,a_2,….a_n are in H.P then (a_1)/(a_2+,a_3,…,a_n),(a_2)/(a_1+a_3+….+a_n),…,(a_n)/(a_1+a_2+….+a_(n-1)) are in