# Harmonic mean

#### Definition:

In sequences and series, the harmonic mean represents a measure of central tendency used to find an average in situations where rates or ratios are involved. It is specifically related to the reciprocal of the arithmetic mean of reciprocals.

#### Formula for Harmonic Mean:

For $n$ numbers ${x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}$ the harmonic mean is calculated as:

Or, more succinctly:

Where:

• $n$ is the number of values.
• ${x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}$ are the individual values.

#### Properties and Usage:

1. Rates and Reciprocals:

• Useful when dealing with rates, speeds, or quantities reciprocally related.
• Example: Average speed in a round trip.
2. Relationship to Arithmetic Mean:

• For $n$ positive real numbers, the harmonic mean is always less than or equal to their arithmetic mean.
3. Balance of Rates:

• Represents the reciprocal of the arithmetic mean of the reciprocals, balancing extreme values.

#### Example:

Consider two speeds $60$ km/h and $40$ km/h. To find their harmonic mean:

So, the harmonic mean of $60$ km/h and $40$ km/h is $48$ km/h.

#### Relationship to Sequences:

• The harmonic mean is related to harmonic progressions (HP), where the reciprocals of terms form an arithmetic progression.