Consider the following scenario:

An accounting firm has increased their client base over three years by the following numbers:

Year 1: +300 000 000 clients

Year 2: +200 000 000 clients

Year 3: +100 000 000 clients

Here, we can use the arithmetic mean to determine the yearly increase of clients:

Arithmetic Mean = $\frac{300 000 000 \+ 200 000 000 \+ 100 000 000}{3}$

Arithmetic Mean = 200 000 000

Therefore, it is fair to say that the company increased their client base by an average of 200 000 000 clients yearly.

Now, consider another accounting firm that has their client base increase information given in percentages:

Year 1: +1.5%

Year 2: +2.0%

Year 3: +2.5%

In this scenario, the annual increases are expressed in relative terms. For example, the number of clients for year $n\+1$ is a ratio of the number of clients for year $n$. The total increase will then depend on the product of these ratios; this number goes into the formula for the geometric mean. Therefore, the geometric mean is a better representation of the average client base increase in this scenario.

Let us illustrate this idea by doing each calculation in turn. If we use the arithmetic mean to calculate the yearly client increase, we would conclude that the accounting firm increased by 2.0% yearly on average. Now, if we consider a company that started with 100 000 000 clients, we would get the following number of clients at the end of the three years:

100 000 000 * 1.020* 1.020 * 1.020 = 106 120 800 clients.

The arithmetic mean does not represent the actual growth. According to the actual figures, the total number of clients at the end of the three years should be:

100 000 000 * 1.015 * 1.020 * 1.025 = 106 118 250 clients.

This is an example of a case where the geometric mean is the appropriate tool to use. The geometric mean for the three years is:

$\sqrt[3]{1.015\cdot 1.020\cdot 1.025}\= 1.01999\.$

Now, calculating the total number of clients based on the geometric mean equates:

100 000 000 * 1.01999* 1.01999 * 1.01999 = 106 118 250 clients.

$\therefore \mathrm{PQ}\=\sqrt{\mathrm{ab}}$