Relation between A.M. ,G.M. and H.M
Definitions:

Arithmetic Mean (A.M.):
 Represents the average of a set of numbers.
 Calculated by summing all values and dividing by the count of values.

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.

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:

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

Equality Conditions:
 The equality in the above inequality holds only when all the numbers in the set are equal.

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

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.): $\text{G.M.}=\sqrt[n]{{x}_{1}\times {x}_{2}\times {x}_{3}\times \dots \times {x}_{n}}$

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.