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Math Help - Arithmetric mean vs. Geometric mean

  1. #1
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    Arithmetric mean vs. Geometric mean

    Show that the geometric mean, radical(ab), is always less than or equal to the arithmetic mean, (a+b)/2.
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  2. #2
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    Hello, Slazenger3!

    Show that the geometric mean, \sqrt{ab}, is always less than or equal to the arithmetic mean, \frac{a+b}{2}

    For any two real numbers, a and b: . (a-b)^2 \:\geq\:0

    We have: . a^2 - 2ab + b^2 \:\geq \:0

    Add 4ab to both sides: . a^2 + 2ab + b^2 \:\geq\:4ab

    Then we have: / (a+b)^2 \:\geq\:4ab

    Divide by 4: . \frac{(a+b)^2}{4} \:\geq \:ab \quad\Rightarrow\quad \left(\frac{a+b}{2}\right)^2 \;\geq\;ab

    Take square root: . \frac{a+b}{2} \:\geq\:\sqrt{ab}


    Therefore, the arithmetic mean is always greater than or equal to the geometric mean.

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  3. #3
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    infact for any n postive numbers, A.M.>=G.M.
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