If `a,A_(1),A_(2),b` are in `A.P., a,G_(1),G_(2),b` are in G.P. and `a, H_(1),H_(2),b` are in H.P., then the value of `G_(1)G_(2)(H_(1)+H_(2))-H_(1)H_(2)(A_(1)+A_(2))` is

Let 2,A_(1),A_(2),14 are in A.P.and 2,H_(1),H_(2),14 are in H.P.,then the value of (H_(1)+H_(2))/(A_(1)+A_(2)) is

Lef a, b be two distinct positive numbers and A and G their arithmetic and geometric means respectively. If a, G_(1),G_(2) , b are in G.P. and a, H_(1),H_(2) , b are in H.P., prove that (G_(1)G_(2))/(H_(1)H_(2))lt (A^(2))/(G^(2)) .

If A_(1),A_(2),G_(1),G_(2) and H_(1),H_(2) be two AMs,GMs and HMs between two quantities then the value of (G_(1)G_(2))/(H_(1)H_(2)) is

Let a,b be positive real numbers.If aA_(1),A_(2),b be are in arithmetic progression a,G_(1),G_(2),b are in geometric progression,and a,H_(1),H_(2),b are in harmonic progression,show that (G_(1)G_(2))/(H_(1)H_(2))=(A_(1)+A_(2))/(H_(1)+H_(2))

If a and b are distinct positive real numbers such that a, a_(1), a_(2), a_(3), a_(4), a_(5), b are in A.P. , a, b_(1), b_(2), b_(3), b_(4), b_(5), b are in G.P. and a, c_(1), c_(2), c_(3), c_(4), c_(5), b are in H.P., then the roots of a_(3)x^(2)+b_(3)x+c_(3)=0 are

if a,a_(1),a_(2),a_(3),......,a_(2n),b are in A.P. and a,g_(1),g_(2),............g_(2n),b are in G.P. and h is H.M. of a,b then (a_(1)+a_(2n))/(g_(1)*g_(2n))+(a_(2)+a_(2n-1))/(g_(2)*g_(2n-1))+.........+(a_(n)+a_(n+1))/(g_(n)*g_(n+1)) is equal

If a,a_(1),a_(2),...,a_(10),b are in A.P.and a,g_(1),g_(2),...,g_(10),b are in G.P.and h is the H.M.betweein a and b,then (a_(1)+a_(2)+...+a_(10))/(g_(1)g_(10))+(a_(2)+a_(3)+...+a_(9))/(g_(2)g_(9))+......+(a_(5)+a_(6))/(g_(5)g_(6))

Let f(x)=(a^(2)+b^(2)-4a-6b+13)(2x^(2)-4x+5),a,b in R such that f(0)=f(1)=f(2). If a,A_(1),A_(2),...A_(10),b is an A.P.and a,H_(1),H_(2)....H_(10),b is a H.P.then find the value of (1)/(10)(sum_(r=4)^(8)A_(r)H_(11-r))

If a_(1),a_(2),...a_(n) are in H.P then the expression a_(1)a_(2)+a_(2)a_(3)+...+a_(n-1)a_(n) is equal to