There is a third type of mean that is used quite often: the harmonic mean. The harmonic mean of numbers a₁, a₂, ..., a_n, none of which is zero, is given by

In a harmonic progression, defined as a progression of the form

each term, except the end ones, is the harmonic mean of its two adjacent terms; that is, if a₁, a₂, ..., a_n are in harmonic progression, then

The proof is left to the reader.
 
 

Monday, June 22, 1998