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Questions and Answers
What is the formula for the geometric mean of n numbers?
What is the formula for the geometric mean of n numbers?
- $(a_{1}a_{2}\cdots a_{n})^{\frac{1}{n}}$
- $(\frac{1}{n}\sum_{i=1}^{n}ln\ a_{i})$
- $(\prod_{i=1}^{n}a_{i})^{\frac{1}{n}}$ (correct)
- $(\frac{1}{n}\sum_{i=1}^{n}a_{i})$
Why are the numbers in the geometric mean formula restricted to being non-negative?
Why are the numbers in the geometric mean formula restricted to being non-negative?
- To simplify the calculation
- To avoid complications related to negative numbers not having real roots (correct)
- To make the formula more intuitive
- To enable the use of logarithms
What is the geometric mean of 2 and 8?
What is the geometric mean of 2 and 8?
- 6
- 5
- 3
- 4 (correct)
How is the geometric mean related to the arithmetic mean in logscale?
How is the geometric mean related to the arithmetic mean in logscale?
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Why are the numbers in the geometric mean formula frequently restricted to being positive?
Why are the numbers in the geometric mean formula frequently restricted to being positive?
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Study Notes
Geometric Mean Formula
- The formula for the geometric mean of n numbers is the nth root of the product of the numbers.
Restrictions on Geometric Mean Formula
- The numbers in the geometric mean formula are restricted to being non-negative because the nth root is undefined for negative numbers.
Example of Geometric Mean
- The geometric mean of 2 and 8 is √(2 × 8) = √16 = 4.
Relationship with Arithmetic Mean
- The geometric mean is related to the arithmetic mean in log scale, meaning that the logarithm of the geometric mean is equal to the arithmetic mean of the logarithms.
Restrictions on Geometric Mean Values
- The numbers in the geometric mean formula are frequently restricted to being positive to avoid complex roots and to ensure a well-defined geometric mean.
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