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 x1,x2,x3,,xn the harmonic mean is calculated as:

Harmonic Mean=n1x1+1x2+1x3++1xn

Or, more succinctly:

Harmonic Mean=ni=1n1xi

Where:

  • n is the number of values.
  • x1,x2,x3,,xn 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:

Harmonic Mean=2160+140=2160+140=2124=48 km/h

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.