# Question: The geometric mean GM is defined as the nth root

The geometric mean (GM) is defined as the nth root of the product of n values. The formula is

GM = n √(X1) (X2) (X3). (Xn)

The geometric mean of 4 and 16 is

GM = √ (4) (16) = √64 = 8

The geometric mean of 1, 3, and 9 is

GM = 3√ (1)(3)(9) = 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.

GM = √ (1.2) (1.1) = 1.1489

GM = √ (120) (110) = 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%.

Raise 1 = 10,000 . 20% = $2000

Raise 2 = 12,000 . 10% = $1200

His total salary raise is $3200. This total is equivalent to

$10,000 . 14.89% = $1489.00

$11,489 . 14.89% = 1710.71 / $3199.71 ≈ $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%. 25.5%

b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. 5.7%

c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. 8.4%

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%. 3.2%

GM = n √(X1) (X2) (X3). (Xn)

The geometric mean of 4 and 16 is

GM = √ (4) (16) = √64 = 8

The geometric mean of 1, 3, and 9 is

GM = 3√ (1)(3)(9) = 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.

GM = √ (1.2) (1.1) = 1.1489

GM = √ (120) (110) = 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%.

Raise 1 = 10,000 . 20% = $2000

Raise 2 = 12,000 . 10% = $1200

His total salary raise is $3200. This total is equivalent to

$10,000 . 14.89% = $1489.00

$11,489 . 14.89% = 1710.71 / $3199.71 ≈ $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%. 25.5%

b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. 5.7%

c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. 8.4%

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%. 3.2%

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