Let a and b be positive real numbers. If a, `A_(1), A_(2)`, b are in arthimatic 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))=((2a+b)(a+2b))/(9ab)`.

Let a ,b be positive real numbers. If a, A_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_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)

Let a ,b be positive real numbers. If a A_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_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)=((2a+b)(a+2b))/(9a b)

If 2, h_(1), h_(2),………,h_(20)6 are in harmonic progression and 2, a_(1),a_(2),……..,a_(20), 6 are in arithmetic progression, then the value of a_(3)h_(18) is equal to

If A_(1),A_(2) are between two numbers, then (A_(1)+A_(2))/(H_(1)+H_(2)) is equal to

If a_(1), a_(2), a_(3) are in arithmetic progression and d is the common diference, then tan^(-1)((d)/(1+a_(1)a_(2)))+tan^(-1)((d)/(1+a_(2)a_(3)))=

Let a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, find an expression for the geometric mean of G_(1), G_(2),...,G_(n) in terms of A_(1), A_(2),...,A_(n), H_(1), H_(2),..., H_(n)

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

Let (a_(1),b_(1)) and (a_(2),b_(2)) are the pair of real numbers such that 10,a,b,ab constitute an arithmetic progression. Then, the value of ((2a_(1)a_(2)+b_(1)b_(2))/(10)) is

Let a_(1),a_(2),a_(3)"……." be an arithmetic progression and b_(1), b_(2), b_(3), "……." be a geometric progression sequence c_(1),c_(2),c_(3,"…." is such that c_(n)= a_(n) + b_(n) AA n in N . Suppose c_(1) = 1, c_(2) = 4, c_(3) = 15 and c_(4) = 2 . The value of sum of sum_(i = 1)^(20) a_(i) is equal to "(a) 480 (b) 770 (c) 960 (d) 1040"

Let a_(1),a_(2),a_(3), . . . be a harmonic progression with a_(1)=5anda_(20)=25 . The least positive integer n for which a_(n)lt0 , is