4.4.4 Geometric and Harmonic Means

Geometric and harmonic means are starting to appear on Number Sense exams in various ways.

Formulas

Geometric Mean

$$G_n = \sqrt[n]{x_1 x_2 \dots x_n}$$

Harmonic Mean

$$H_n = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}$$

Example: Geometric Mean

What is the geometric mean of 6, 4, and 9?

$$\sqrt[3]{6 \times 4 \times 9} = \sqrt[3]{216} = 6$$

Harmonic Mean of Polynomial Roots

For roots $r, s, t$ of a cubic polynomial:

$$H_3 = \frac{3rst}{rs + rt + st}$$

Example: What is the harmonic mean of the roots of $x^3 + 2x^2 - 3x + 7 = 0$?

$$H_3 = \frac{3 \cdot (-7)}{-3} = 7$$

Dual-Labor Problem

The classic “two people working together” problem uses the harmonic mean:

$$\text{Time together} = \frac{1}{2} \times H_2 = \frac{1}{\frac{1}{t_1} + \frac{1}{t_2}}$$

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Problem Set 4.4.4

Practice these problems. Type your answer and press Enter to check: