## 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 by where is the common difference. We add to each term to get the next term. This means that so 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 terms The 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 by where is the common ratio. We multiply each term by to get the next term. This means that and so 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 of numbers. 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.
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