Harmonic Mean questions This is kinda a stupid question but I have seen harmonic mean expressed so many different ways to the point where I only know that the harmonic mean of 2 numbers is $\frac{2xy}{x+y}$ So what if I wanted the harmonic mean of more than 2 numbers? and can I take the harmonic mean of a positive number and a negative number? 

Harmonic mean of $x_1,\ldots,x_n$ is defined as
$$
\frac{n}{x_1^{-1}+x_2^{-1}+\ldots+x_n^{-1}}
$$
Usually, you would want all $x_i$ to be positive, so that the denominator is guaranteed not to be zero. In case of $n=2$ you get, like you said,
$$
\frac{2}{x_1^{-1}+x_2^{-1}}=\frac{2}{\frac{x_2}{x_1x_2}+\frac{x_1}{x_1x_2}}=\frac{2x_1x_2}{x_1+x_2}.
$$
It is a special case of a power or generalised mean. A power mean of positive numbers $x_1,\ldots,x_n$ with exponent $\mu$, where $\mu$ is a real number, is defined as
$$
\left( \frac{x_1^\mu+x_2^\mu+\ldots+x_n^\mu}{n}\right)^{1/\mu}
$$
For $\mu=1$ you get the arithmetic mean, for $\mu=-1$ you get the harmonic mean. It can be shown that the mean is monotone in $\mu$, and the limit as $\mu\to 0$ is the geometric mean, while the limits at $\mu=+\infty,-\infty$ are, respectively, the maximum and the minimum.
In fact, there are also continuous and weighted analogues of these, and similar facts are true.
