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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.
The arithmetic mean is a value that is derived by dividing the total of a set of values by the number of items in the set. The geometric mean is a value between the first and last of a set of values organized in a geometric progression. Compare median, mode. Statistics.