The geometric mean (GM) is defined as the nth root of the product of n values. The formula is
$$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$
The geometric mean of 4 and 16 is
$$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$
The geometric mean of $1,3,$ and 9 is
$$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$
The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown.
$$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$
or
$$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$
His salary is $120 \%$ at the end of the first year and $110 \%$ at the end of the second year. This is equivalent to an average of $14.89 \%,$ since $114.89 \%-100 \%=$ $14.89 \% .$
$$\begin{array}{l}{\text { This answer can also be shown by assuming that }} \\ {\text { the person makes } \$ 10,000 \text { to start and receives two }} \\ {\text { raises of } 20 \% \text { and } 10 \% .}\end{array}$$
$$\begin{array}{l}{\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000} \\ {\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200}\end{array}$$
$$\text { His total salary raise is } \$ 3200 . \text { This total is equivalent to }$$
$$\begin{array}{l}{\$ 10,000 \cdot 14.89 \%=\$ 1489.00} \\ {\$ 11,489 \cdot 14.89 \%=\$ 1710.71}\end{array}$$
$${\$ 3199.71} \approx \$ 3200$$
Find the geometric mean of each of these.
a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%.
b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%.
c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%.
d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.