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Arithmetic is strictly dealing with real numbers and operators, (addition, subtraction etc.) Mathermatics covers a much broader description including arithmetic, calculus, algebra, etc... You can say that Arithmetic is a subset of Mathematics.
Arithmetic mean is easy to use, quick to calculate and can be useful when trying to find the average for many things in life. However, it is an inappropriate metric to use to determine the actual average return of an investment. The geometric mean is a more difficult metric to use and understand.