Harmonic Progression: Overview, Questions, Preparation

Sequence and Series 2021 ( Sequence and Series )

Rachit Kumar Saxena

Rachit Kumar SaxenaManager-Editorial

Updated on Jun 28, 2021 10:33 IST

What is the Harmonic Progression?

When you arrange a series of numbers into a predictable pattern, it is termed progression. Progression is a sequence of numbers that follow particular rules. Progression is can be of three types, i.e. Arithmetic Progression, Geometric Progression & Harmonic Progression. 

For solving the problems related to harmonic progression, firstly perceive the corresponding sum of an arithmetic progression. In simple terms, the nth term of the progression must be equal to the nth terms reciprocal of the related Arithmetic Progression. 

Harmonic Progression Formula

Harmonic Progression’s nth term= 1/ [a+(n-1)d]


“a” depicts the first-term of the Arithmetic Progression.

“d” represents the common difference.

“n” depicts overall no. of terms in Arithmetic Progression.

Harmonic Progression Sum:

If there is a harmonic expression, i.e. 1/a, 1/a+d, 1/a+2d, …., 1/a+(n-1), the actual formula for finding the sum of n terms is:


In which,

“a” is first term

“d” is difference

“ln”: natural logarithm

Illustrative Examples of Harmonic Progression

To understand the working of harmonic progression, let’s go through some solved examples: 

1. Determine the 4th and 8th term of the harmonic progression 6, 4, 3,…


Harmonic progression = 6, 4, 3

Find the corresponding arithmetic progression for the H.P.

Arithmetic Progression = ⅙, ¼, ⅓, ….

By A.P. T2 -T1 is equal to T3 -T2 
i.e. 1/12 = d

For finding the 4th term of the given A. P, the formula is: 

nth term = a+(n-1)d

Where, a = ⅙ and d= 1/12

For 4th term n=4

Putting these values in the formula:

4th term = (⅙) +(4-1)(1/12)

=> (⅙)+(3/12)

=> 5/12

Repeat the same steps for 8th term

8th = (⅙) +(8-1)(1/12)

= (⅙)+(7/12)

= 9/12

As we know, harmonic progression is reciprocal, I.e. reverse of an Arithmetic Progression:

Thus, the 4th term becomes 1/4th term = 12/5 and

8th term becomes 1/8th term = 12/9 or 4/3

2. If the first two terms of a harmonic progression are 1/16 and 1/13, find the maximum partial sum?


Harmonic series will be: 1/16, 1/13, 1/10,1/7, 1/4, 1/1, 1/-2, 1/-5

Maximum partial sum for this series will be 1/16 + 1/13 + 1/10 +1/7 + 1/4 + 1/1 = 1.63

3. Compute the 16th term of H.P. if the 6th and 11th term of H.P. is 10 and 18.


Write H.P. in terms of A.P.:

6th term of Arithmetic Progression= a+5d = 1/10 [Equation 1]

11th term of Arithmetic Progression= a+10d = 1/18 [Equation 2]

Solve equation 1 and 2 for getting the values of a and d:

a =13/90

d = -2/ 225

For finding 16th term, write expression as:


= (13/90) – (2/15) 

= 1/90

The 16th term of the Harmonic Progression is equal to 1/16th term of an Arithmetic Progression that is 90

Thus, the 16th term is 90.

FAQ’s on Harmonic Progression

Q: What is the formula of Harmonic Mean?

A: Harmonic Mean can be calculated as
 n /[(1/a) + (1/b)+ (1/c)+(1/d)+….]
In which,
a, b, c, d represents the values of the progression n depicts the total no of values present in the progression.

Q: What are the four types of sequences?

A: The four types of Sequences and series are Arithmetic Sequences, Geometric Sequences, Harmonic Sequences and 
Fibonacci Numbers.

Q: Is harmonic progression in JEE mains?

A: No, questions related to harmonic progression aren’t asked in JEE Mains, whereas you can expect related questions in JEE Advanced.

Q; Why do we calculate the harmonic mean?

A: We calculate harmonic mean because it helps us find the multiplicative and divisor relationships amongst the given fractions without thinking about the common denominators. 

Q; How do you find the common difference in harmonic progression?

A: You can find the common difference by only subtracting two- adjacent terms. You can find every term by recursively adding d, i.e. common difference within the preceding term.

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