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Math Help - Harmonics

  1. #1
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    Harmonics

    Hello friends!

    Question is:
    The harmonic mean of 2 numbers, a and b is
    ____2____
    _1_ + _1_
    a______b

    The harmonic mean of 3 numbers, a, b and c, is
    ______3______
    _1_ + _1_ + _1_
    a______b____c

    a) Simplify the expressions for 2 and 3 numbers and find a simplified expression for the harmonic mean of 4 numbers.

    b) Find the harmonic mean of the numbers 2, 3, 4 7 and 9.
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  2. #2
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    Quote Originally Posted by Bajan View Post
    Hello friends!

    Question is:
    The harmonic mean of 2 numbers, a and b is
    ____2____
    _1_ + _1_
    a______b

    The harmonic mean of 3 numbers, a, b and c, is
    ______3______
    _1_ + _1_ + _1_
    a______b____c

    a) Simplify the expressions for 2 and 3 numbers and find a simplified expression for the harmonic mean of 4 numbers.
    Where is the difficulty?
    \frac{2}{\frac{1}{a} + \frac{1}{b}}

    = \frac{2}{\frac{1}{a} + \frac{1}{b}} \cdot \frac{ab}{ab}

    = \frac{2ab}{\left ( \frac{1}{a} + \frac{1}{b} \right ) ab}

    = \frac{2ab}{b + a}

    Use a similar procedure for the other cases.

    -Dan
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  3. #3
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    Hello, Bajan!

    \text{The harmonic mean of two numbers, }a\text{ and }b\text{ is: }\:\frac{2}{\frac{1}{a} + \frac{1}{b}}

    \text{The harmonic mean of three numbers, }a, b\text{ and }c\text{, is: }\;\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}


    (a) Simplify the expressions for 2 and 3 numbers
    and find a simplified expression for the harmonic mean of 4 numbers.

    \text{Two numbers: }\;\frac{2}{\frac{1}{a}+\frac{1}{b}} \;=\;\frac{2ab}{a + b}

    \text{Three numbers: }\;\frac{3}{\frac{1}{a} + \frac{1}{b}+\frac{1}{c}} \;=\;\frac{3abc}{ab + bc + ac}

    \text{Four numbers: }\;\frac{4}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\f  rac{1}{d}} \;=\;\frac{4abcd}{abc + abd + acd + bcd}



    (b) Find the harmonic mean of the numbers 2, 3, 4, 7 and 9.
    \text{Five numberse: }\;\frac{5}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c} + \frac{1}{d} + \frac{1}{e}} \;=\;\frac{5abcde}{abcd + abce + abde + acde + bcde}


    H.M. (2,3,4,7,9) . =\;\frac{5(2)(3)(4)(7)(9)}{(2)(3)(4)(7) + (2)(3)(4)(9) + (2)(3)(7)(9) + (2)(4)(7)(9) + (3)(4)(7)(9)}

    . . . . . . . . . . . . = \;\;\frac{7560}{2022} \;\;=\;\;\frac{1260}{337}

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  4. #4
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    Thanks guys! I couldn't get my head around the idea but now it's seems to make sense.
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