Media aritmetică, geometrică și armonică

\begin{align*}
m_a=\bar{x}&=\frac{x_1+x_2+\ldots+x_n}{n}\\[1em]
(x_1-\bar{x})&+(x_2-\bar{x})+\\[1em]+&\ldots+(x_n-\bar{x})=0\\[1em]
\overline{x+a}&=\overline{x}+a\\[1em]
\overline{c\cdot x} &=c\cdot\overline{x}\\[1em]
m_{ap}&=\frac{p_1 x_1+p_2 x_2 + \ldots + p_n x_n}{p_1+p_2+\ldots + p_n}
\end{align*}

m_g=\sqrt[n]{x_1\cdot x_2 \cdot \ldots \cdot x_n}\\[1em]
m_g=(x_1\cdot x_2 \cdot \ldots \cdot x_n)^\frac{1}{n}\\[1em]
m_g=e^{\frac{1}{n}\cdot(ln(x_1)+ln(x_2)+\ldots+ln(x_n))}\\[1em]
\exists\, 1\le i \le n, x_i =0\implies m_g=0

m_h=\frac{n}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}\\[1em]
m_h = \frac{1}{m_a(\frac{1}{x_1},\ldots,\frac{1}{x_n}) }

\forall \, 1\le i \le n, x_i \gt 0 \implies \\[1em]min(x_1,\ldots,x_n)\le m_h\le m_g \le m_a \le max(x_1,\ldots,x_n)