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Math Help - arithmetic mean and geometric mean

  1. #1
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    arithmetic mean and geometric mean

    If LNy is the arithmetic mean of LNx and LNz show that y is the geometric mean of x and z.

    I found that xz=LN(xz) hence xz=1=y.
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  2. #2
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    You missed that.
    \ln(y)=\dfrac{\ln(x)+\ln(z)}{2}
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  3. #3
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    I misread the question as if it said "If y ...". I will probably manage it now.
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  4. #4
    MHF Contributor harish21's Avatar
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    Use what Plato has stated in his post above to get y = \sqrt{(xz)} and recall the definition of geometric mean.
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  5. #5
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    LNy=0.5(LNx+LNz)
    If x,y and z are in G.P. then (xz)^2=y.
    LN(xz)^2=0.5(LNx+LNz)
    =0.5(LN(xz))
    =LN(xz)^2
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  6. #6
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    Quote Originally Posted by Stuck Man View Post
    LNy=0.5(LNx+LNz)
    If x,y and z are in G.P. then (xz)^2=y.
    LN(xz)^2=0.5(LNx+LNz)
    =0.5(LN(xz))
    =LN(xz)^2
    You've got squares where you should have square roots.
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  7. #7
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Stuck Man View Post
    LNy=0.5(LNx+LNz)
    If x,y and z are in G.P. then (xz)^2=y.
    LN(xz)^2=0.5(LNx+LNz)
    =0.5(LN(xz))
    =LN(xz)^2
    0.5(ln(xz)) = ln(xz)^{(1/2)} = ln(\sqrt{(xz)})
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