# Geometric Mean

The **geometric mean** is a type of power mean. For a collection \(\{a_1, a_2, \cdots, a_n\}\) of positive real numbers, their geometric mean is defined to be \[\text{GM}(a_1, \cdots, a_n) := \sqrt[n]{a_1 a_2 \cdots a_n}.\] For instance, the geometric mean of \(4\) and \(9\) is \[\text{GM}(4,9) = \sqrt{4\cdot 9} = \sqrt{36} = 6\] and the geometric mean of \(8\), \(27\), and \(64\) is \[\text{GM}(8, 27, 64) = \sqrt[3]{8\cdot 27\cdot 64} = 2\cdot 3 \cdot 4 = 24.\]

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## Geometric Interpretation

The terminology "*geometric* mean" comes from the fact that this quantity has a simple geometric interpretation. Consider the below picture, where \(AH = a\) and \(HC = b\).

From the Pythagorean Theorem, one obtains the three equations \[AH^2 + BH^2 = AB^2\] \[HC^2 + BH^2 = BC^2\] \[AB^2 + BC^2 = AC^2\] This implies \[AH^2 + 2 BH^2 + HC^2 = AC^2\] and substituting \(AH = a\), \(HC = b\) gives \[a^2 + b^2 + 2BH^2 = (a+b)^2 = a^2 + b^2 + 2ab \implies BH = \sqrt{ab}.\] Thus, \(BH = \text{GM}(AH, HC)\).

In particular, note that the radius of the pictured semicircle is \((a+b)/2\), and the radius must be greater than or equal to \(BH\) (where equality occurs only when \(B\) is directly above the semicircle's center). Hence, \[\frac{a+b}{2} \ge \sqrt{ab}\] for all positive real numbers \(a\) and \(b\). This is the two-variable case of the AM-GM inequality.

## Properties

Observe that by taking logarithms, we have \[ \log \text{GM}(a_1, \cdots, a_n) = \log \sqrt[n]{a_1 a_2 \cdots a_n} = (\log a_1 + \log a_2 + \cdots \log a_n)/n \] Hence, the logarithm of the geometric mean is the arithmetic mean of the logarithms.

## Problems

Suppose \(\{a_1, \ldots, a_{99}\}\) is a geometric sequence. If \(a_{49} = 18\) and \(a_{51} = 8\), what is \(\text{GM}(a_1, \ldots, a_{99})\)?