上文題要：
Part 1 我們利用Backward induction 證明了算幾不等式，現在我們利用算幾不等式，證明Geometric mean (G.M.) 幾何平均值大於Harmonic mean (H.M.) 調和平均值
\[there\ are\ n\ numbers,\ x_1,\ x_2,\ ....\ ,\ x_n\ ,\ \ \forall i,\ 1\le i\le n,\ x_i\ge0\]
Arithmetic mean (A.M.)算術平均值
\[A.M.\ =\frac{x_1+x_2+...\ +x_n}{n}\]
Geometric mean (G.M.) 幾何平均值
\[G.M.\ =\left(x_1x_2...x_n\right)^{\frac{1}{n}}\]
Harmonic mean (H.M.) 調和平均值
\[H.M.=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}\]
\[A.M\ge G.M.\ge H.M\]
Part 2
\[證明\ G.M\ge H.M.\]
\[H.M.=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}\le\frac{n}{n\left(\frac{1}{x_1x_2...x_n}\right)^{\frac{1}{n}}}=\left(x_1x_2...x_n\right)^{\frac{1}{n}}=G.M.\ \ \left(By\ A.M.\ge G.M.\right)\]
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