Geometric and Harmonic Means

This section covers two types of means that appear on tests.

• Geometric Mean ($G_n$): The nth root of the product of n numbers: $G_n = \sqrt[n]{x_1x_2...x_n}$.
• Harmonic Mean ($H_n$): The reciprocal of the arithmetic mean of the reciprocals: $H_n = \frac{n}{\frac{1}{x_1} + \dots + \frac{1}{x_n}}$.

Problems may involve finding the harmonic mean of the roots of a cubic, which relates to the product and sum of roots taken two at a time. Another common application is the dual-labor problem (e.g., two people working together), where the combined time is related to the harmonic mean of their individual times.