To derive the relationship between the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) for a set of positive numbers, we can use algebraic reasoning and properties. Let’s assume we have a set of n positive numbers: x₁, x₂, x₃, …, xₙ.
1. Arithmetic Mean (AM):
The Arithmetic Mean is calculated by summing all the values in the set and dividing by the total count, n.
AM = (x₁ + x₂ + x₃ + … + xₙ) / n
2. Geometric Mean (GM):
The Geometric Mean is calculated by taking the nth root of the product of all the values in the set.
GM = (x₁ * x₂ * x₃ * … * xₙ)^(1/n)
3. Harmonic Mean (HM):
The Harmonic Mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of all the values in the set.
HM = n / ((1/x₁) + (1/x₂) + (1/x₃) + … + (1/xₙ))
To derive the relationship between these means, we can start by considering the following inequality:
AM ≥ GM
Now, substitute the formulas for AM and GM:
(x₁ + x₂ + x₃ + … + xₙ) / n ≥ (x₁ * x₂ * x₃ * … * xₙ)^(1/n)
Raise both sides to the power of n:
(x₁ + x₂ + x₃ + … + xₙ) ≥ (x₁ * x₂ * x₃ * … * xₙ)
This inequality states that the sum of the values is greater than or equal to the product of the values.
Now, consider the following inequality:
HM ≥ GM
Substitute the formulas for HM and GM:
n / ((1/x₁) + (1/x₂) + (1/x₃) + … + (1/xₙ)) ≥ (x₁ * x₂ * x₃ * … * xₙ)^(1/n)
Raise both sides to the power of n:
n ≥ ((1/x₁) + (1/x₂) + (1/x₃) + … + (1/xₙ))^n
This inequality states that n is greater than or equal to the reciprocal of the arithmetic mean of the reciprocals of the values, raised to the power of n.
Thus, we have derived the relationships:
AM ≥ GM and HM ≥ GM
These inequalities show that the Arithmetic Mean is greater than or equal to the Geometric Mean, and the Harmonic Mean is greater than or equal to the Geometric Mean for a set of positive numbers.