Harmonic Progression – Meaning, Formulas & Examples

A progression is a series of numbers arranged in a predictable pattern. It is a type of number set that follows specific, definite rules. Many types of sequences (also called progressions) and series are studied in mathematics such as arithmetic, geometric, harmonic, etc.

Let’s understand what harmonic progression is and what the different formulas used to solve problems are based on it.

What is a Harmonic Progression?

A harmonic progression is formed by taking the reciprocal of the terms of the arithmetic progression. The sequence $x_1, x_2, x_3, x_4, …, x_n$ will be in harmonic progression if $\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \frac{1}{x4}, …, \frac{1}{x_n}$ are in arithmetic progression.

You know that an A.P. is written as $a$, $a + d$, $a + 2d$, $a + 3d$, …, $a + (n – 1)d$, where $a$ is the first term, $d$ is common difference and $n$ is the total number of terms. Then $\frac{1}{a}$, $\frac{1}{a + d}$, $\frac{1}{a + 2d}$, $\frac{1}{a + 3d}$, …, $\frac{1}{a + (n – 1) d}$ forms a harmonic progression. Both $a$ and $d$ have non-zero values.

harmonic progression

Examples of Harmonic Progression

Example 1: The progression $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, … is a harmonic progression.

Taking the reciprocals of the individual terms of the sequence $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, …, we get $1$, $2$, $3$, $4$, …, which is an arithmetic progression where first term $a = 1$ and the common difference $d = 2 – 1 = 3 – 2 = 4 – 3 = … = 1$.

Example 2: The progression $\frac{1}{3}$, $\frac{1}{10}$, $\frac{1}{17}$, $\frac{1}{24}$, … is a harmonic progression.

Taking the reciprocals of the individual terms of the sequence $\frac{1}{3}$, $\frac{1}{10}$, $\frac{1}{17}$, $\frac{1}{24}$, …, we get $3$, $10$, $17$, $24$, …, which is an arithmetic progression where first term $a = 3$ and the common difference $d = 10 – 3 = 17 – 10 = 24 – 17 = … = 7$.

Harmonic Progression Formulas

The harmonic progression formulas help us to solve problems on harmonic progression quickly and easily. The different harmonic progression formulas are

  • $n^{th}$ Term of Harmonic Progression
  • Harmonic Mean
  • Harmonic Progression Sum

$n^{th}$ Term of Harmonic Progression

$n^{th}$ term of harmonic progression is the reciprocal of the nth term of the arithmetic progression. The $n^{th}$ term of the harmonic progression is the reciprocal of the sum of the first term and the $(n – 1)$ times the common difference. 

$n^{th}$ term of H.P. = $\frac{1}{a + (n – 1)d}$ where $a$ is the first term, $d$ is the common difference of the corresponding A.P.

Examples of $n^{th}$ Term of Harmonic Progression

Example 1: Find the $16^{th}$ term and the $n^{th}$ term of the harmonic progression $\frac{1}{5}$, $\frac{1}{11}$, $\frac{1}{17}$, $\frac{1}{23}$, …

Given harmonic progression is $\frac{1}{5}$, $\frac{1}{11}$, $\frac{1}{17}$, $\frac{1}{23}$, …

The corresponding arithmetic progression is $5$, $11$, $17$, $23$, …

First term $a = 5$

Common difference $d = 11 – 5 = 17 – 11 = 23 – 17 = 6$

$n^{th}$ term of the A.P. = $a + (n – 1)d$

Therefore $16^{th}$ term of the A.P. = $5 + (16 – 1) \times 6$

$= 5 + 15 \times 6$

$= 5 + 90 = 95$

Thus $16^{th}$ term of the H.P. = $\frac{1}{95}$.

And $n^{th}$ term of the H.P. = $\frac{1}{a + (n – 1)d} = \frac{1}{5 + (n – 1) \times 6} = \frac{1}{5 +6 (n – 1)}$

Example 2: Find the $12^{th}$ term of the harmonic progression, if the fifth term is $\frac{1}{16}$, and the eighth term is $\frac{1}{25}$.

Let the first term and the common difference of the corresponding A.P. be $a$ and $d$ respectively.

The fifth term of an H.P. = $\frac{1}{16}$

$=>$ the fifth term of corresponding A.P. = $16$ 

$=> a + (5 – 1)d = 16 => a + 4d  = 16$ ——————————– (1)

The eighth term of an H.P. = $\frac{1}{25}$

$=>$ the eighth term of corresponding A.P. = $25$ 

$=> a + (8 – 1)d = 25 => a + 7d  = 25$ ——————————– (2)

Solving (1) and (2), we get $a = 4$ and $d = 3$

$12^{th}$ term of an H.P. = $\frac{1}{12^{th} \text{ term of A.P.}}$

$= \frac{1}{a + (12 – 1)d}$

$= \frac{1}{4 + 11 \times 3}$

$= \frac{1}{4 + 33} = \frac{1}{37}$

Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The formula to calculate the harmonic mean of $n$ values $x_1$, $x_2$, $x_3$, …, $x_n$ is given by Harmonic Mean =$\frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + … + \frac{1}{x_n}}$. 

Harmonic Mean of Two Values

If $a$ and $b$ are two values then the harmonic mean of $a$ and $b$ is given by $\frac{2}{\frac{1}{a} + \frac{1}{b}}$

$= \frac{2}{\frac{a + b}{ab}} = \frac{2ab}{a + b}$

Examples on Harmonic Mean

Example 1: Find the harmonic mean of $2$ and $8$.

Harmonic mean of two numbers $a$ and $b$ is given by $\frac{2ab}{a + b}$

Therefore harmonic mean of $2$ and $8$ is  $\frac{2 \times 2 \times 8}{2 + 8} = \frac{32}{10} = \frac{16}{5} = 3.2$

Example 2: Find the harmonic mean of $1$, $5$, and $8$

Harmonic Mean =$\frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + … + \frac{1}{x_n}}$. 

Here $n = 3$, $x_1 = 1$, $x_2 = 5$, and $x_3 = 8$

Therefore harmonic mean = $\frac{3}{\frac{1}{1} + \frac{1}{5} + \frac{1}{8}}$

$= \frac{3}{\frac{40 + 8 + 5}{40}}$

$= \frac{3}{\frac{53}{40}}$

$= \frac{3 \times 40}{53} = \frac{120}{53}$

Harmonic Progression Sum

The sum of n terms in a harmonic progression can be determined easily if the first term and the value of n terms are known. 

If the terms $\frac{1}{a}$, $\frac{1}{a + d}$, $\frac{1}{a + 2d}$, $\frac{1}{a + 3d}$, …, $\frac{1}{a + 3d}$ make a harmonic progression, the formula to find the sum of $n$ terms in the harmonic progression is obtained by the formula $S_n = \frac{1}{d} \ln \frac{2a + (2n – 1)d}{2a – d}$

where 

$a$ is the first term of the corresponding A.P.

$d$ is the common difference of the corresponding A.P.

$\ln$ is the natural logarithm

Examples on Harmonic Progression Sum

Example 1: Find the sum of first $15$ terms of the harmonic progression $\frac{1}{5}$, $\frac{1}{11}$, $\frac{1}{17}$, $\frac{1}{23}$, …

The given H.P. is $\frac{1}{5}$, $\frac{1}{11}$, $\frac{1}{17}$, $\frac{1}{23}$, …

The corresponding A.P. is $5$, $11$, $17$, $23$, …

Therefore, $a = 5$, $d = 11 – 5 = 6$, and $n = 15$

The sum of the first $15$ terms of the H.P. is $S_{15} = \frac{1}{6} \ln \frac{2 \times 5 + (2 \times 15 – 1) \times 6}{2 \times 5 – 6}$

$= \frac{1}{6} \ln \frac{10 + (30 – 1) \times 6}{10- 6}$

$= \frac{1}{6} \ln \frac{10 + 29\times 6}{4}$

$= \frac{1}{6} \ln \frac{10 + 174}{4}$

$= \frac{1}{6} \ln \frac{184}{4}$

$= \frac{1}{6}(\ln 184 – \ln 4)$

$= \frac{1}{6} \times (5.2149 – 1.3863)$

$= \frac{1}{6} \times 3.8286 = 0.6381$

Properties of Harmonic Progression

The following are the properties of harmonic progression:

  • No term of H.P. can be zero.
  • If H is the H.M. between a and b, then we have the following properties:
    • $\frac{1}{\text{H} – a} + \frac{1}{\text{H} – b} = \frac{1}{a} + \frac{1}{b}$
    • $(\text{H} – 2a)(\text{H} – 2b) = \text{H}^2$
    • $\frac{\text{H} + a}{\text{H} – a} + \frac{\text{H} + b}{\text{H} – b} = 2$
  • If $a$ and $b$ are two positive real numbers, then the relation between A.M., G.M., and H.M. is $\text{A. M.} \times \text{H. M.} = \text{G.M.}^2$

Uses of Harmonic Progression

Some of the important applications of harmonic progression are

  • The average speed of a vehicle across two sets of equal distances can be computed using the harmonic mean of the respective speeds. If the speed of the vehicle is $x$ km/h for the first $d$ kilometers and it is $y$ km/h for the next $d$ kilometers, then the average speed of the vehicle across the entire distance is equal to the harmonic mean of these two speeds. Average Speed = $\frac{2xy}{x + y}$.
  • The density of a mix of substances or the density of the alloy of two or more substances of equal weight and equal weight percentage composition can be computed using the harmonic mean of the densities of the individual components.
  • The focal length of a lens is equal to the harmonic mean of the distance of the object($u$) from the lens, and the distance of the image($v$) from the lens. $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.
  • In geometry, the radius of the incircle of a triangle is equal to one-third of the harmonic mean of the altitudes of the triangle.
  • In the field of finance, the profit-earning ratio is computed using the concept of the weighted harmonic mean of individual components.
  • Scientists use harmonic formulas to determine the result of their experiments.
  • Harmonic sequencing is used in the field of music to practice notes.
  • The formula of harmonic progression is applied to determine the degree to which water boils when its temperature is modified by the same value each time.
  • Businesses and large corporations also employ harmonic progression to efficiently run their functions such as sale predictions, financial budgeting, weather forecasting, etc.

Practice Problems

  1. What is a harmonic progression?
  2. What is the relation between harmonic progression and arithmetic progression?
  3. Check whether the following are in harmonic progression or not.
    1. $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, ….
    2. $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{6}$, $\frac{1}{8}$, ….
    3. $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, …
    4. $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, …
  4. If the sum of reciprocals of the first $11$ terms of an HP series is $110$, find the $6^{th}$ term of HP.
  5. Find the $4^{th}$ and $8^{th}$ term of the series $6, 4, 3,$ …
  6. The $2^{nd}$ term of an HP is $\frac{40}{9}$ and the $5^{th}$ term is $\frac{20}{3}$. Find the maximum possible number of terms in H.P.
  7. If the first two terms of a harmonic progression are $\frac{1}{16}$ and $\frac{1}{13}$, find the maximum partial sum.
  8. The second term of an H.P. is $\frac{3}{14}$ and the fifth term is  $\frac{1}{10}$. Find the sum of its $6^{th}$ and the $7^{th}$ term.
  9. If the sixth term of an H.P. is $10$ and the $11^{th}$ term is $18$ Find the $16^{th}$ term.

FAQs

What is the general formula of HP?

Harmonic progression is a sequence of numbers in which the reciprocal of each term in the sequence is in arithmetic progression. If the terms $x_1, x_2, x_3, x_4, …, x_n$  are in H.P., then $\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \frac{1}{x4}, …, \frac{1}{x_n}$ are in A.P.

What is harmonic progression called?

A sequence of numbers where the reciprocals of the terms are in the arithmetic progression is called a harmonic progression.

What is a real-life example of harmonic progression?

harmonic progression

The leaning tower of lire is a perfect example of a harmonic progression. A cluster of cubes having identical sides is placed over each other to get as much sideways as possible or sideways distance.

Why is it called a harmonic?

The harmonic sequence is so named because it is exactly the sequence of points on a taut string that deliver musical “harmonics” when the string is touched there as it is plucked.

How is harmonic mean used in daily life?

Harmonic means are often used in averaging things like rates (e.g., the average travel speed given a duration of several trips).

Conclusion

A harmonic progression is formed by taking the reciprocal of the terms of the arithmetic progression. The sequence $x_1, x_2, x_3, x_4, …, x_n$ will be in harmonic progression if $\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \frac{1}{x4}, …, \frac{1}{x_n}$ are in arithmetic progression. The harmonic progression finds applications in business, finance, and solving scientific and engineering problems.

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