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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

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


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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