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More on means

The notions of arithmetic and geometric means can be extended to more than two numbers. For example, if $a,b,c$ are positive numbers, their arithmetic and geometric means are defined to be

$\dfrac{a+b+c}{3} \qquad\text{and}\qquad \sqrt[3]{abc},$

respectively. In general, if $a_1,a_2,\dots,a_n$ are $n$ positive real numbers, their arithmetic and geometric means are

$\dfrac{a_1 + a_2 + \dots + a_n}{n} \qquad\text{and}\qquad \sqrt[n]{a_1a_2\dots a_n},$

respectively.

Exercise 14

Suppose that $a_1,a_2,\dots,a_n$ is a sequence of positive real numbers and $b > 1$ . Show that the logarithm (base $b$ ) of the geometric mean of these numbers is equal to the arithmetic mean of the numbers $\log_b a_1,\ \log_b a_2,\ \dots,\ \log_b a_n$ .

The harmonic mean $H$ of two positive numbers $a$ and $b$ is defined by

$\dfrac{1}{H} = \dfrac{1}{2}\Biggl( \dfrac{1}{a} + \dfrac{1}{b}\Biggr).$

It is the reciprocal of the average of the reciprocals of $a$ and $b$ . We can rearrange this equation as

$H = \dfrac{2ab}{a+b}.$

More generally, the harmonic mean $H$ of positive numbers $a_1,a_2,\dots,a_n$ is defined by

$\dfrac{1}{H} = \dfrac{1}{n}\Biggl(\dfrac{1}{a_1} + \dfrac{1}{a_2} + \dots + \dfrac{1}{a_n}\Biggr).$

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