there are some geometric means between 8 and 1/8. if the first mean to the last mean is 16:1 , find the number of means
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mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1, x2, ..., xn, the geometric mean is defined as
{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}} {\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, {\displaystyle {\sqrt {2\cdot 8}}=4} {\displaystyle {\sqrt {2\cdot 8}}=4}. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, {\displaystyle {\sqrt[{3}]{4\cdot 1\cdot 1/32}}=1/2} {\displaystyle {\sqrt[{3}]{4\cdot 1\cdot 1/32}}=1/2}.