**Contents**:

Harmonic mean

Harmonic progression

## What is the Harmonic Mean?

The harmonic mean is a very specific type of average. It’s generally used when dealing with averages of units, like speed or other rates and ratios.

The formula is:

If the formula above looks daunting, all you need to do to solve it is:

- Add the reciprocals of the numbers in the set.
- Divide the number of items in the set by your answer to Step 1.

Not that great with calculating reciprocals? Try this online calculator.

## Examples

Watch the video or read on below for a few examples:

What is the harmonic mean of 1, 5, 8, 10?

- Add the reciprocals of the numbers in the set: 1/1 + 1/5 + 1/8 + 1/10 = 1.425
- Divide the number of items in the set by your answer to Step 1. There are 4 items in the set, so:

4 / 1.425 = 2.80702

**Tip**: Check your calculations with this online calculator.

## Difference Between the Harmonic Mean and Arithmetic Mean

The difference between the two means is a little bit of a mind-bender, so if you don’t quite get the concept at first, you aren’t alone. I can remember having to read the explanation several times and doing some examples on paper to assure myself it was correct.

Here is an example of a problem involving (the more common) **arithmetic mean**:

- Joe drives a car at 20 mph for the first hour and 30 mph for the second. What’s his average speed?
- The average speed is 20 + 30 / 2 = 25 mph.

And now, the same example with the **harmonic mean**:

- Joe drives a car at 20 mph for the half of the journey and 30 mph for the second half. What’s his average speed?
- For this problem note that we’re being told he went a certain speed for a journey segment. We need the harmonic mean:
- = 2/(1/20 + 1/30)
- = 2(0.05 + 0.033)
- = 2 / 0.083
- = 24.09624 mph.

The difference between these two is that the first problem is calculating an **average speed based on time**, while the second is based on **distance**. You’ll notice that the harmonic mean is slightly less than the arithmetic mean. This is always the case, that the harmonic mean will be the lowest average.

## Harmonic Progression

A **harmonic progression **is a sequence of numbers where each term is the harmonic mean of the neighboring terms. A “progression” is just a sequence of numbers that follows a pattern.

A harmonic progression takes the form:

In this formula:

- a is non-zero and
- -a/d cannot be a natural numbers.

## Alternate Definition

You can also make this sequence with reciprocals of an arithmetic progression (an *arithmetic progression* is a series of numbers where each successive term has a constant added to it). More specifically, if the following sequence is an arithmetic progression:

Then it is also a harmonic progression, as long as the numbers are non-zero.

## Properties of the Harmonic Progression

One interesting property of the harmonic progression is that, unless a = 1 and k = 0, it can never sum up to an integer. The reason for this is that at least one of the denominators of the progression will be divisible by a prime number that doesn’t divide any of the other denominators.

## Examples of Harmonic Progressions

Let’s say *a* and *d* are both ½. Then the harmonic progression will equal:

**12, 6, 4, 3, 12/5, 2…n. **

The *n*th term will always be 12/(1 + n).

Suppose a was 1/10 and d was – 2/30. Then the terms of the progression would equal:

**10, 30, -30, -10, -6, -30/7…n. **

The *n*th term will always be 10/(1 – 2n/3).

The graph below shows part of the harmonic progression where a_{n} = 1 / n. For this series, a and d are both 1. The first term, 1/n, is 1, the second term is 1/(1 + 1) =2, and so on.

Sometimes, this progression is referred to as “The” harmonic series.

## References

Joyce, D. Introduction to Series and Sequences. Math 221 Calculus. Published Spring 2013. Retrieved from https://www2.clarku.edu/faculty/djoyce/ma121/seqser.pdf on March 30, 2019.

Zenth, Tommy “The Sum of the Harmonic Progression as an Integral.” From Wolfram Demonstrations Project, Published online March 7, 2011. Retrieved from http://demonstrations.wolfram.com/TheSumOfTheHarmonicProgressionAsAnIntegral/ on March 30, 2019.

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**Stephanie Glen**. "Harmonic Mean / Progression: Definition, Examples" From

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