Harmonic Progression
A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression.
Definition:
If the sequence:
$$
a_1, a_2, a_3, \dots
$$
is a harmonic progression, then:
$$
\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots
$$
forms an arithmetic progression.
General Term of HP:
If the reciprocal sequence follows an AP:
$$
\frac{1}{a_n} = \frac{1}{a} + (n - 1)d
$$
Then the nth term of the HP is:
$$
a_n = \frac{1}{\frac{1}{a} + (n - 1)d}
$$
Harmonic Mean (HM):
The harmonic mean of two numbers $a$ and $b$ is given by:
$$
\text{HM} = \frac{2ab}{a + b}
$$
The harmonic mean is useful in problems involving average speeds, resistances in parallel circuits, and rates.
Important Notes:
- The terms of an HP are always positive if all terms of the AP formed by reciprocals are defined.
- Unlike AP or GP, you rarely sum the terms of an HP — the focus is usually on identifying or inserting harmonic means.
- To insert harmonic means between two numbers, insert arithmetic means between their reciprocals, then take the reciprocal of each result.
Relationship Between Harmonic, Geometric, and Arithmetic Means
Given any two positive numbers $a$ and $b$, we can define the following:
- Arithmetic Mean (AM): $\displaystyle \text{AM} = \frac{a + b}{2}$
- Geometric Mean (GM): $\displaystyle \text{GM} = \sqrt{ab}$
- Harmonic Mean (HM): $\displaystyle \text{HM} = \frac{2ab}{a + b}$
Key Relationship (Inequality):
$$
\text{HM} \le \text{GM} \le \text{AM}
$$
Equality holds only when $a = b$.
Why This Matters:
- AM is useful for equal sharing (e.g., average grade).
- GM is used when growth or proportion is involved (e.g., interest rates, scaling).
- HM applies when the quantity being averaged is a rate (e.g., average speed, resistance).
Fun Observation:
For any two positive numbers $a$ and $b$, the GM is always exactly the geometric mean between the HM and AM:
$$
\text{GM}^2 = \text{AM} \cdot \text{HM}
$$
This elegant identity reinforces the symmetrical and interconnected nature of the three means.
