Arithmetic and Geometric Mean

The terms 'arithmetic mean' and 'geometric mean' are derived naturally from the definition of arithmetic series and geometric series respectively.

Successive terms in an arithmetic sequence are defined bywhereis the common difference. We addto each term to get the next term. This means thatso that

Any term is the arithmetic mean of the term immediately preceding and the term immediately succeeding that term.

We can generalise this to any odd number of termsThe middle term is the arithmetic mean of the terms immediately preceding and the terms immediately succeeding:

In turn this can be generalized in the obvious way to an even number of terms.

Successive terms in an geometric sequence are defined bywhereis the common ratio. We multiply each term byto get the next term. This means thatandso that

From this we obtain

This equation says that any term is the geometric mean of the term immediately preceding and the term immediately succeeding that term.

We can generalise this to any odd number of terms,The middle term is the geometric mean of the terms immediately preceding and the terms immediately succeeding:

In turn this can be generalized in the obvious way to an even number of terms.

In fact a sequence can be any list ofnumbers. The arithmetic mean is just the sum of all the terms divided by the number of terms and the geometric mean is just the nth root of the product of the terms.