What Is Harmonic Progression? Definition, Formula & Examples

A harmonic progression is a sequence of nonzero real numbers whose reciprocals form an arithmetic progression. This concept underlies many advanced results in algebra and is essential for connecting the ideas of arithmetic mean, geometric mean, and harmonic mean.

Formal Definition of Harmonic Progression and the General Term

Let $\{h_n\}$ be a sequence of real numbers. The sequence is said to be a harmonic progression if each $h_n \neq 0$ for all $n$, and the sequence $\left\{\dfrac{1}{h_n}\right\}$ forms an arithmetic progression. In particular, there exist constants $a \in \mathbb{R}$ and $d \in \mathbb{R} \setminus \{0\}$ such that for all $n \geq 1$, \[ \dfrac{1}{h_n} = a + (n-1)d \]

Solving for $h_n$ gives the general (or $n$-th) term of a harmonic progression: \[ h_n = \dfrac{1}{a + (n-1)d} \] where $a$ is the first term and $d$ is the common difference of the associated arithmetic progression of reciprocals. Note that $h_n \neq 0$ for any finite $n$ as long as $a + (n-1)d \neq 0$.

Relation Between Harmonic Progression and Arithmetic Progression

If $\{h_n\}$ is a harmonic progression, then the sequence $\{x_n\}$ defined by $x_n = \dfrac{1}{h_n}$ is an arithmetic progression. Conversely, for any arithmetic progression $\{x_n\}$ with $x_n = a + (n-1)d$, the sequence $h_n = 1/x_n$ is a harmonic progression provided $x_n \neq 0$ for all $n$.

Harmonic Mean Between Two Numbers and Three-Term Harmonic Progressions

If $a$ and $b$ are two nonzero real numbers, the harmonic mean $H$ of $a$ and $b$ is defined as \[ H = \dfrac{2ab}{a + b} \]

Given three numbers $a$, $H$, $b$, these are in harmonic progression if and only if $H$ is the harmonic mean of $a$ and $b$ as defined above. That is, $a$, $H$, $b$ form a harmonic progression if and only if \[ H = \dfrac{2ab}{a + b} \] because their reciprocals $1/a$, $1/H$, $1/b$ form an arithmetic progression. This can be verified stepwise as follows:

Consider the reciprocals $x = 1/a$, $y = 1/H$, $z = 1/b$. For $x$, $y$, $z$ to be in arithmetic progression, the middle term must satisfy: \[ 2y = x + z \] Substituting: \[ 2 \cdot \frac{1}{H} = \frac{1}{a} + \frac{1}{b} \] \[ \frac{2}{H} = \frac{a + b}{ab} \] Multiplying both sides by $H$: \[ 2 = \frac{a + b}{ab} \cdot H \] \[ 2ab = (a + b)H \] Dividing both sides by $(a + b)$: \[ H = \frac{2ab}{a + b} \]

General Harmonic Mean of Multiple Numbers

If $x_1, x_2, \dots, x_n$ are $n$ nonzero real numbers, the harmonic mean $H$ of the $n$ numbers is defined as \[ H = \frac{n}{\displaystyle\sum_{i=1}^{n} \frac{1}{x_i}} \] This definition arises because if these $n$ numbers are in harmonic progression, their reciprocals form an arithmetic progression, and this mean averages their reciprocals arithmetically, taking the reciprocal at the end.

For the special case $n = 3$, given numbers $a$, $b$, $c$ (all nonzero), their harmonic mean is \[ H = \frac{3}{\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}} = \frac{3abc}{ab + bc + ca} \]

Sum of the First $n$ Terms of a Harmonic Progression

There is no general closed formula for the sum of the first $n$ terms of a harmonic progression analogous to the sum formula for arithmetic or geometric progressions. For the specific case where the harmonic progression is given by \[ h_k = \frac{1}{a + (k-1)d}, \quad k = 1, 2, \ldots, n \] the sum is given by \[ S_n = \sum_{k=1}^{n} \frac{1}{a + (k-1)d} \] which, when $d \neq 0$, can be represented using properties of logarithms and integrals. An explicit formula (for $a > 0, d > 0$) is \[ S_n = \frac{1}{d} \left[ \ln \left( \frac{a + nd - d}{a} \right) \right] \] The derivation uses properties of telescoping sums and the observation that \[ \int \frac{1}{a + x d}\,dx = \frac{1}{d} \ln |a + x d| + C \] Therefore, the sum is a difference of logarithms: \[ S_n = \frac{1}{d} \ln \left( \frac{a + nd - d}{a} \right) \]

Relationship Between Arithmetic Mean, Geometric Mean, and Harmonic Mean

If $p$ and $q$ are two positive real numbers, the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are defined as: \[ AM = \frac{p + q}{2}, \qquad GM = \sqrt{p q}, \qquad HM = \frac{2 p q}{p + q} \]

Among any two positive real numbers, the following inequality always holds: \[ AM \geq GM \geq HM \] Equality holds if and only if $p = q$.

There is also a direct algebraic relationship among the means: \[ GM^2 = AM \cdot HM \] This can be verified by substitution as follows: \[ GM^2 = (\sqrt{p q})^2 = p q \] \[ AM \cdot HM = \frac{p + q}{2} \cdot \frac{2 p q}{p + q} = p q \]

For comprehensive understanding of relationships between means and progressions, refer to Arithmetic, Geometric, and Harmonic Progression.

Criterion for Three Numbers to be in Harmonic Progression

Let $a$, $b$, $c$ be three nonzero numbers. These numbers are in harmonic progression if and only if their reciprocals are in arithmetic progression: \[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \] Multiplying both sides by $b$: \[ 2 = \frac{b}{a} + \frac{b}{c} \] Multiply both sides by $ac$: \[ 2 a c = b c + b a \] \[ 2 a c = b(a + c) \] Divide both sides by $(a + c)$: \[ b = \frac{2 a c}{a + c} \] Thus, the middle term $b$ of any harmonic progression is the harmonic mean of its neighboring terms $a$ and $c$.

Illustrative Examples in Harmonic Progression

Example 1: Find the fifth term of the harmonic progression $6,\ 4,\ 3,\ \ldots.$

First, take the reciprocals to obtain the corresponding arithmetic progression: \[ \frac{1}{6},\quad \frac{1}{4},\quad \frac{1}{3},\ \ldots \]

Compute the common difference $d$: \[ \frac{1}{4} - \frac{1}{6} = \frac{3 - 2}{12} = \frac{1}{12} \] \[ \frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12} \] Thus, $d = \frac{1}{12}$.

The general term of the arithmetic progression is \[ a_n = \frac{1}{6} + (n - 1)\left(\frac{1}{12}\right) \]

For $n = 5$, \[ a_5 = \frac{1}{6} + 4 \left(\frac{1}{12}\right) = \frac{1}{6} + \frac{1}{3} = \frac{1 + 2}{6} = \frac{3}{6} = \frac{1}{2} \]

Thus, the fifth term in the harmonic progression is the reciprocal of $a_5$: \[ h_5 = \frac{1}{a_5} = \frac{1}{1/2} = 2 \]

Example 2: For which value of $p$ do $5,\, p,\, 10$ form a harmonic progression?

Take reciprocals: $\dfrac{1}{5},\ \dfrac{1}{p},\ \dfrac{1}{10}$ must form an arithmetic progression. Thus, \[ 2 \cdot \frac{1}{p} = \frac{1}{5} + \frac{1}{10} \] \[ 2 \cdot \frac{1}{p} = \frac{2 + 1}{10} = \frac{3}{10} \] \[ \frac{1}{p} = \frac{3}{20} \] \[ p = \frac{20}{3} \]

Application in Triangle: Harmonic Progression and Triangle Sides

Let the altitudes of a triangle be in arithmetic progression. Denote the sides by $a$, $b$, $c$ and corresponding altitudes by $p$, $q$, $r$. The area $\Delta$ can be expressed as \[ \Delta = \frac{1}{2} a p = \frac{1}{2} b q = \frac{1}{2} c r \] Thus, $a p = b q = c r = k$ (say, for some constant $k$).

Express the sides in terms of altitudes: \[ a = \frac{k}{p},\quad b = \frac{k}{q},\quad c = \frac{k}{r} \] If $p$, $q$, $r$ are in arithmetic progression, then $a, b, c$ (being reciprocals) are in harmonic progression.

Standard Results Involving Harmonic Progression

If $a$, $b$, $c$ are in geometric progression, then $b^2 = a c$. If $a$, $b$, $c$ are in harmonic progression, then $2b = \dfrac{2 a c}{a + c}$. The reciprocal sequence of a geometric progression forms a geometric progression, and the reciprocal of an arithmetic progression forms a harmonic progression (if all terms are nonzero).

Necessary Restrictions and Observations on Harmonic Progressions

In any harmonic progression, no term can be zero. The corresponding arithmetic progression of reciprocals would otherwise be undefined for that term. Harmonic sequences generally do not permit closed forms for arbitrary sums, except in special telescoping or limiting scenarios.

Summary of Harmonic Progression for Examinations

The foundation for questions on harmonic progression lies in converting the sequence into its reciprocal and solving via corresponding arithmetic progression principles. Typical exam questions involve identification, calculation of general terms, verification of harmonic progression, relationship among means, or application to geometric scenarios such as triangles.