The geometric mean (GM) is defined as the \(n\) th 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 \(114.89
\%-100 \%=\) \(14.89 \% .\)
This answer can also be shown by assuming that the person makes \(\$ 10,000\) to
start and receives two raises of \(20 \%\) and \(10 \%\).
$$\begin{array}{l}\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000 \\\\\text {
Raise } 2=12,000 \cdot 10 \%=\$ 1200\end{array}$$
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 \(5.5 \% .\)
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