# Relation between A.M. ,G.M. and H.M

#### Definitions:

1. Arithmetic Mean (A.M.):

• Represents the average of a set of numbers.
• Calculated by summing all values and dividing by the count of values.
2. Geometric Mean (G.M.):

• Represents the $n$th root of the product of $n$ numbers.
• Calculated by multiplying all values and taking the $n$th root, where $n$ is the number of values.
3. Harmonic Mean (H.M.):

• Represents the reciprocal of the average of reciprocals of a set of numbers.
• Calculated by taking the reciprocal of each value, calculating the arithmetic mean of these reciprocals, and then taking the reciprocal of the result.

#### Relationships:

1. Inequality Relationship:

• For a set of positive real numbers, the following inequality holds: $\text{A.M.}\ge \text{G.M.}\ge \text{H.M.}$
2. Equality Conditions:

• The equality in the above inequality holds only when all the numbers in the set are equal.
3. Special Case - Two Numbers:

• For two positive real numbers $a$ and $b$, the relationship among A.M., G.M., and H.M. is: $\text{A.M.}\ge \text{G.M.}\ge \text{H.M.}$
4. Special Case - Three Numbers:

• For three positive real numbers $a$, $b$, and $c$, the relationship among A.M., G.M., and H.M. is: $\text{A.M.}\ge \text{G.M.}\ge \text{H.M.}$

#### Formulas:

For a set of $n$positive real numbers ${x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}$:

• Arithmetic Mean (A.M.): $\text{A.M.}=\frac{{x}_{1}+{x}_{2}+{x}_{3}+\dots +{x}_{n}}{n}$

• Geometric Mean (G.M.): $\text{G.M.}=\sqrt[n]{{x}_{1}×{x}_{2}×{x}_{3}×\dots ×{x}_{n}}$

• Harmonic Mean (H.M.): $\text{H.M.}=\frac{n}{\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}}+\dots +\frac{1}{{x}_{n}}}$

#### Significance:

• A.M., G.M., and H.M. are measures of central tendency used in different scenarios, emphasizing different aspects of a dataset.
• A.M. provides a balance between values.
• G.M. emphasizes the effect of multiplicative factors.
• H.M. highlights reciprocal relationships.

#### Applications:

• Finance: G.M. for compound interest rates, A.M. for averaging expenses, H.M. for averaging speeds.
• Physics: G.M. in wave frequencies, A.M. in balancing forces, H.M. in calculating resistances.