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Arithmetic mean

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The arithmetic mean of a set of numbers (or variables) is the sum of all the numbers, divided by the number of numbers - the average of the set. If we let denote Arithmetic Mean,

AM=\frac{x_1+x_2+\cdots+x_n}{n}

is the arithmetic mean of the numbers \displaystyle x_1,x_2,\ldots,x_n.

For example, if I wanted to find the average of the numbers 3, 1, 4, 1, and 5, I would compute:

\frac{3+1+4+1+5}{5} = \frac{14}{5}.


Arithmetic means show up frequently in contest problems, often in the AM-GM inequality or its variant, the RMS-AM-GM-HM inequality.

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