Question

The geometric mean of a set values lies between arithmetic mean and​

Answer 1

Step-by-step explanation:

Calculation. The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, an intersection of the two which always lies in between

Answer 2

Answer:

Arithmetic mean

In mathematics, the arithmetic mean is a number that can be obtained by dividing the sum of the values ​​in a group by the number of values ​​in that group.

Geometric Mean

With respect to any given number of values ​​and any number of observations, the geometric mean is defined as the nth root of the product of those values ​​and those observations.

Explanation:

Arithmetic mean

In mathematics, the arithmetic mean is a number that can be obtained by dividing the sum of the values ​​in a group by the number of values ​​in that group. If a1, a2, a3,...,an is a number of sets of values ​​or the arithmetic progression, then the following statements are true:

AM=(a₁+a₂+a₃+….,+aₙn)/n

Geometric Mean

With respect to any given number of values ​​and any number of observations, the geometric mean is defined as the nth root of the product of those values ​​and those observations.

GM = n√(a₁+a₂+a₃+….,+aₙ)

or

GM = (a₁+a₂+a₃+….,+aₙ)1 ⁄ n

The relation between Arithmetic Mean and Geometric Mean

The relation between Arithmetic Mean and Geometric Mean is given below:

The following properties are:

Property I: The Arithmetic Means of two positive numbers can never be less than their Geometric Mean.

Property II: If A be the Arithmetic Means and G be the Geometric Means between two positive numbers m and n, then the quadratic equation whose roots are m, n is x^2 - 2Ax + G^2 = 0.

Property III: If A be the Arithmetic Means and G be the Geometric Means between two positive numbers, then the numbers are A ± √A^2 - G^2.

Property IV: If the Arithmetic Mean of two numbers x and y is to their Geometric Mean as p : q, then, x : y = (p + √(p^2 - q^2) : (p - √(p^2 - q^2).

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