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Last updated on July 5th, 2025

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Harmonic Progression

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Harmonic progression is the reciprocal of arithmetic progression. This progression does not contain any terms with zero. It is used in real-life situations such as financial calculations and problems related to speed, sound, and rates. In this article, we will discuss its definition, formulas, examples, and applications.

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What is Harmonic Progression?

A harmonic progression is formed by taking the reciprocals of an arithmetic progression. In arithmetic progression, we add the same number each time, like a (a + d), (a + 2d), (a + 3d),. In a harmonic progression, we take the reciprocal of each term in an arithmetic progression, so we will get 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d).

 

 

Here, 
a = first term, 
d = common difference between the terms. 
To make sure the terms are defined and the sequence progresses, so ‘a’ and ‘d’ can never be zero. This pattern continues indefinitely, making it an infinite sequence. 

 

 

Let’s understand this clearly using the following example:
Rahul rides a bicycle on a straight road. If he rides at different speeds over different parts of the journey, his travel time follows a harmonic pattern.
If he travels at a speed of 10 km/h, 20 km/h, 30 km/h, 40 km/h,..., these speeds are an arithmetic progression because he adds 10 km/h each time.
But if he calculates the time taken for each part of the journey, the values will be 1/10, 1/20, 1/30, 1/40,..., which forms a harmonic progression.

 

 

Harmonic progression is used in average speed calculations, especially when the distances are covered at different speeds and the total time follows a harmonic pattern.
 

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Relationship between AM, GM and HM

The relationship among Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) is: 
AMGMHM

 


This means that the Arithmetic Mean is always the largest, followed by the Geometric Mean, and the Harmonic Mean is the smallest. 

 


To understand this, consider any two numbers a and b. The formulas for the arithmetic mean, geometric mean, and harmonic mean are as follows. The arithmetic mean is the average of two numbers.

 


AM = a+b2

 


The geometric mean is equal to the square roots of the product of two numbers. 

 


GM = ab

 


The harmonic mean of two numbers is equal to the inverse of the average of their reciprocals. 

 


HM = 2aba+b

 

The formula for the relation between AM, GM, and HM is the product of the arithmetic and harmonic mean is equal to the square of the geometric mean.

 


AM × HM = GM2

 

By deriving this formula, we are able to identify it better.


AM × HM = a+b2 × 2aba+b
    = ab


Here, ab can be derived as ab2 = GM2
 

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What is the Formula for nth Term of Harmonic Progression

Harmonic progression has some important formulas that help in calculations. These formulas help us find specific terms, the average value, and the total sum of a harmonic sequence. 

 

The nth term in HP is the reciprocal of the nth term of an arithmetic progression. The formula for HP is:

 

Hn = 1a + (n - 1) . d

Where, a is the first term of arithmetic sequence.
d is the difference between the terms.
n is the position of the term to be determined.

Harmonic Mean: It is the type of average used in harmonic progression. It is useful when dealing with speed, distance, or rates. For any two numbers a and b, the harmonic mean is: 
HM = 2aba + b
For any three numbers, the harmonic mean is:

HM = 3abab + bc + ca

Finding the total sum of an HP is more complicated than arithmetic or geometric progressions as it involves fractions. The sum of first n terms is given by:

Sn = 1d log (2a + (2n - 1)d2a - d)

This formula is used in more advanced calculations, especially in science and engineering.
 

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Real Life Applications of Harmonic Progression

Harmonic Progression is an important concept that can be applied to real-life situations where quantities are inversely related. Here are a few real-life applications of Harmonic Progression.

 

 

Speed and Travel Time:
When a vehicle moves at different speeds for equal distances, the total time taken follows a harmonic progression. This concept is used in aviation, shipping, and railway scheduling to calculate travel times accurately. For example: The harmonic mean is used to calculate the total time when a car travels 100 km at 50 km/h and another 100 km at 100 km/h. This helps in accurately planning the total travel time.

 

 

Electrical Circuits:
When the resistors are connected parallel, the total resistance follows a harmonic progression. Each term is always a reciprocal of the resistance. It also improves the efficiency of electrical circuits, power grids and transformers. 

 

 

Finance and Economics:
Harmonic mean is used in finance for averaging values when dealing with ratios, like interest rates, stock price calculations, and investment returns. This is useful in funds, risk analysis, and portfolio management. For example: Suppose that a stock analyst is comparing two stocks with P/E ratios of 10 and 20. The harmonic mean is used in place of the standard average:
HM = (2 × 10 × 20)/ (10 + 20) = 400/30 =13.333
 

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Common Mistakes and How To Avoid Them in Harmonic Progression

Harmonic progression involves working with reciprocals and fractions, which can sometimes cause mistakes. Here are some common mistakes and tips on how to avoid them.
 

Mistake 1

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Forgetting to take reciprocal.
 

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Students often make the mistakes like not taking the reciprocal when converting Arithmetic progression to Harmonic progression. Always remember that an HP is formed by taking the reciprocal of an arithmetic sequence. For example, if AP = 2, 5, 8,11,..., then HP is ½, ⅕, ⅛, 1/11,...
 

Mistake 2

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Confusing harmonic mean and arithmetic mean
 

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Students mistakenly use the arithmetic mean formula instead of the harmonic mean formula when calculating averages. Arithmetic mean is calculated as: AM =a + b2 and harmonic mean can be calculated as: HM = 2aba + b.
 

Mistake 3

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Ignoring zero or negative terms
 

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Using a = 0 or a negative value for d in HP formulas leads to an invalid sequence. Ensure that a is not 0 and d should be a positive number so that the terms in the HP follow a logical sequence.
 

Mistake 4

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Misusing the nth term
 

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Some students apply the nth term formula of an AP directly to HP without taking the reciprocal. Always apply the reciprocal of the AP formula. The nth term of an HP can be expressed as:
Hn= 1/ [AP nth term]
 

Mistake 5

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Misplacing terms in a formula
 

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When applying the formula for the nth term, students mix up the values for a and d.  Always check whether a is the first term of the A.P and d is the difference between two consecutive terms in the sequence.
 

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Solved Examples of Harmonic Progression

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Problem 1

The 2nd and 4th terms of a harmonic progression are 2 and 4. Find the 5th term.

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 8
 

Explanation

In H.P., take reciprocals to form an A.P.
Reciprocals: 1/2 and 1/4 (these are the 2nd and 4th terms of the A.P.)
Let’s find the common difference (d):
 Difference between positions = 4–2 = 2
 So, d = (1/4–1/2) / 2 = (-1/4) / 2 = -1/8
Now find the 5th A.P. term:
2nd term = 1/2
3rd term = 1/2 + (-1/8) = 3/8
4th term = 3/8 + (-1/8) = 1/4
5th term = 1/4 + (-1/8) = 1/8
So, 5th H.P. term = Reciprocal of 1/8 = 8
 

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Problem 2

In a harmonic progression, the 1st term is 1 and the 3rd term is 1/3. Find the 2nd term.

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 ½
 

Explanation

In an H.P, take the  reciprocals to form an AP:
 1st H.P. term = 1 → reciprocal = 1 
 3rd  H.P. term = 1/3   → reciprocal = 3
 So, in A.P., Middle (2nd) term = average of 1st and 3rd terms:
(1 + 3)/2 = 2 → 4/2 = 2
Taking the reciprocal of 2→ 1/2.
 

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Problem 3

The 3rd and 5th terms of a harmonic progression are 3 and 6. Find the 4th term.

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 4
 

Explanation

We first take reciprocals to form an A.P: 
3rd H.P. term = 3 → reciprocal = 1/3
5th H.P. term = 6 → reciprocal = 1/6
Now, we find the 4th term:
4th term of A.P. = average of 3rd and 5th:
 = (1/3 + 1/6)/2 = (1/2)/2 = 1/4 
Take the reciprocal of 1/4 = 4.
 

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Problem 4

In a harmonic progression, the 2nd term is 5 and the 5th term is 2. Find the 3rd term.

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 10/3 or 3.33
 

Explanation

 Reciprocals give an A.P: 
2nd H.P. term = 5 →  reciprocal = 1/5
5th H.P. term = 2 →  reciprocal =  1/2 
Common difference = (1/2–1/5)/3 = (5–2)/10 ÷ 3 = 3/10 ÷ 3 = 1/10
To find the 3rd A.P. term:
= 1/5 + 1/10 = 2/10 + 1/10 = 3/10
Take reciprocal of 3/10 → H.P. term = 10/3 or approx. 3.33
 

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Problem 5

The 1st and 4th terms of a harmonic progression are 10 and 5. Find the 2nd term.

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7.5
 

Explanation

 Reciprocals form A.P.:
1st H.P. term = 10 → reciprocal = 1/10
4th H.P. term = 5 → reciprocal = 1/5
Steps between 1st and 4th = 3
 Common difference (d):
 d = (1/5–1/10) / 3 = (2–1)/10 ÷ 3 = 1/10 ÷ 3 = 1/30
Now, we find the 2nd A.P. term = 1st + d = 1/10 + 1/30 = (3 + 1)/30 = 4/30 = 2/15
 reciprocal of 2/15 = 15/2 = 7.5
 

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FAQs of Harmonic Progression

1. Is every HP also a GP?

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2.How is harmonic progression used in Physics?

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3.What happens if all terms in an AP are equal?

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4.Where is Harmonic Progression used in real life?

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5.What is the formula for the nth term of a Harmonic Progression?

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6.How can children in Indonesia use numbers in everyday life to understand Harmonic Progression?

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7.What are some fun ways kids in Indonesia can practice Harmonic Progression with numbers?

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8.What role do numbers and Harmonic Progression play in helping children in Indonesia develop problem-solving skills?

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9.How can families in Indonesia create number-rich environments to improve Harmonic Progression skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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