# The Means

As is customary, group a finite sequence of numbers a_{1}, a_{2}, ..., a_{n} in a vector form **a** = (a_{1}, a_{2}, ..., a_{n}). The arithmetic and harmonic means of the sequence defined as

A(**a**) = (∑**a**)/n = (∑a_{i})/n and H(**a**) = n/(∑**1/a**) = n/(∑1/a_{i}),

where the summation is being carried from i = 1 through i = n. Observe that H(**a**) = 1/A(**1/a**). This leads to a more general definition

M_{r}(**a**) = (∑**a**^{r}/n)^{1/r} = (∑a_{i}^{r}/n)^{1/r}

where I assume that all a_{i}'s are positive and r is a real number different from 0. For example, A(**a**) = M_{1}(**a**) whereas H(**a**) = M_{-1}(**a**).

In general, M_{r}(**a**) is the *mean* value of numbers a_{1}, a_{2}, ..., a_{n} with the exponent r. M_{2}(**a**) is known as the *quadratic* average.

For r = 0, the following complements the definition of M_{r}(**a**):

M_{0}(**a**) = (a_{1}a_{2} ... a_{n})^{1/n}

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