Inequality of arithmetic and geometric means

$Inequality of$ $arithmetic and geometric means$

${\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[ {n}]{x_{1}\cdot x_{2}\cdots x_{n}}}\,,$

$x 1 = x 2 = \cdot \cdot \cdot = x n .$

$x+y / 2$ $≧ \sqrt x y$

$x+y+z / 3$ $≧ 3 \sqrt x yz$

$n=2k$

$x n$ $+y n$ $≧ 2 \sqrt x n y n$ $≧ n \sqrt x y$

$n=2k-1$

$x n$ $+y n$ $+z n ≧$ $3\sqrt x n y n z$ $≧ n \sqrt x yz$

${\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[ {n}]{x_{1}\cdot x_{2}\cdots x_{n}}}\,,$

$x 1 = x 2 = \cdot \cdot \cdot = x n .$

$T.theodor 2022.8.14$