Pythagorean means

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In this activity we explore the three different means of the ancient Greeks.

The arithmetic mean

The arithmetic mean is the good-old mean that we are all familiar with.

What is the mean that we are all familiar with? Explain how to compute the mean of $a_1,a_2,\dots ,a_n$ . Give some examples.

The geometric mean

The geometric mean is a bit different. The geometric mean of $a_1,a_2,\dots , a_n$ is given by: $\left (\prod _{i=1}^n a_i\right )^{1/n}$

Explain an analogy between the arithmetic mean and the geometric mean.

Can you explain the geometric mean in terms of geometry? First do it for 2 numbers. Next do it for three.

The harmonic mean

The harmonic mean might be the most mysterious of all. The harmonic mean of $a_1,a_2,\dots , a_n$ is given by: $\frac {n}{\sum _{i=1}^n \frac {1}{a_i}}$

Can you find a connection between the harmonic mean and music?

In the United States, the fuel efficiency of a car is usually given in the units: $\frac {\text {miles}}{\text {gallon}}$ However, in Europe, the fuel efficiency of a car is usually given in the units: $\frac {\text {liters}}{100 \mathrm {km}}$ Give some examples of fuel efficiency (both efficient and inefficient) with each set of units.

Now suppose that a car gets $60\frac {\text {miles}}{\text {gallon}}$ and another car gets $20\frac {\text {miles}}{\text {gallon}}$ . What is the average fuel efficiency?

Now suppose that a car gets $4\frac {\text {liters}}{100 \mathrm {km}}$ and another car gets $20\frac {\text {liters}}{100 \mathrm {km}}$ . What is the average fuel efficiency?

Compare your answers to the last two questions. Something fishy is going on, what is it?