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Three harmonic means `H_(1),H_(2),H_(3)` are inserted between the two numbers 20 and 4. Then find `H_(1),H_(2)` and `H_(3)`.

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If 8 harmonic means are inserted between two numbes a and b (altb) such that arithmetic mean of a and b is 5/4 times equal to geometric mean of a and b then ((H_(8)-a)/(b-H_(8))) is equal to

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)

Knowledge Check

  • Two A.M.'s A_(1) and A_(2) , two G.M.'s G_(1) and G_(2) and two H.M's H_(1) and H_(2) are inserted between any two numbers, then H_(1)^(-1) + H_(2)^(-1) equals

    A
    `A_(1)^(-1) + A_(2)^(-1)`
    B
    `G_(1)^(-1) + G_(2)^(-1)`
    C
    `(G_(1)G_(2))/(A_(1) + A_(2))`
    D
    `(A_(1) + A_(2))/(G_(1)G_(2))`

  • Two AMs. A_1 and A_2 , two GMs. G_1 and G_2 and two HMs. H_1 and H_2 are inserted between any two numbers. Then find the arithmetic mean between H_1 and H_2 in terms of A_1, A_2, G_1, G_2 .

    A
    `(A_1 + A_2)/(2G_1G_2)`
    B
    `(A_1 - A_2)/(2G_1 G_2)`
    C
    `(A_1 + A_2)/(2 + G_1G_2)`
    D
    `(G_1 + G_2)/(2A_1A_2)`

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

    A
    `(H_(1)H_(2))/(G_(1)G_(2))`
    B
    `(G_(1)G_(2))/(H_(1)H_(2))`
    C
    `(H_(1)H_(2))/(A_(1)A_(2))`
    D
    `(G_(1)G_(2))/(A_(1)A_(2))`

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