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 nth root of the product of n numbers.
    • Calculated by multiplying all values and taking the nth 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: A.M.G.M.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: A.M.G.M.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: A.M.G.M.H.M.

Formulas:

For a set of npositive real numbers x1,x2,x3,,xn:

  • Arithmetic Mean (A.M.): A.M.=x1+x2+x3++xnn

  • Geometric Mean (G.M.): G.M.=x1×x2×x3××xnn

  • Harmonic Mean (H.M.): H.M.=n1x1+1x2+1x3++1xn

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.