# Category:Pythagorean Means

This category contains results about **Pythagorean Means**.

Definitions specific to this category can be found in Definitions/Pythagorean Means.

The **Pythagorean means** are as follows:

### Arithmetic Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers.

The **arithmetic mean** of $x_1, x_2, \ldots, x_n$ is defined as:

- $\ds A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$

That is, to find out the **arithmetic mean** of a set of numbers, add them all up and divide by how many there are.

### Geometric Mean

Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers.

The **geometric mean** of $x_1, x_2, \ldots, x_n$ is defined as:

- $\ds G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$

### Harmonic Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all strictly positive.

The **harmonic mean** $H_n$ of $x_1, x_2, \ldots, x_n$ is defined as:

- $\ds \dfrac 1 {H_n} := \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$

That is, to find the **harmonic mean** of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals.

## Subcategories

This category has the following 3 subcategories, out of 3 total.