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Table of contents
1.
Introduction 
2.
Harmonic Progression
2.1.
Common Difference
2.2.
General Term
2.3.
Examples
3.
Sum of n terms in an H.P.
3.1.
Examples
4.
Applications of Harmonic Progression
5.
FAQ
6.
Key Takeaways
Last Updated: Mar 27, 2024

Harmonic Progression

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Introduction 

Progression is defined as a series that exhibits a specific pattern. It is also known as a sequence. Every (n+1)th term in a progression is obtained by applying a specific rule on its nth term. For example, 5, 10, 15, 20, ... is a progression because every number is obtained by adding 5 to its previous number. This pattern may vary from progression to progression.

In mathematics, there are mainly three types of progressions: 

  1. Arithmetic progressions
  2. Geometric progressions
  3. Harmonic progressions
     

This blog will discuss one of the types of progression,i.e., Harmonic Progression.

Harmonic Progression

A Harmonic Progression is defined as a sequence of real numbers determined by taking the reciprocal of an arithmetic sequence with non-zero elements.

For example, given an arithmetic progression 3, 6, 9, 12, ... then the harmonic progression of this progression can be written as 1/3, 1/6, 1/9, 1/12, …

Consider some of the following examples to understand H.P.

  1. 1/4, 1/7, 1/10, 1/13, ...
  2. 1, 1/3, 1/5, 1/7, …
  3. 1/5, 1/10, 1/15, 1/20, …

 

Harmonic Mean: Harmonic mean of a progression is defined as the reciprocal of the arithmetic mean of the reciprocals of any given progression.

For example, if 1, 2, 3, 4, … represents an A.P., then 1, 1/2, 1/3, 1/4,... will represent the respective H.P. Here, the arithmetic mean of the A.P. = (1+2+3+...)/n.

Hence, the harmonic mean will be = n/(1+1/2+1/3+...)

Common Difference

The common difference is the difference between any two consecutive numbers of that sequence. The common difference of an H.P can be calculated using the formula d= an - an-1.
Consider the following examples:

  1. Given a sequence: 1, 1/2, 1/4, 1/6, …
    The common difference of the given sequence is d= 1/4 - 1/2 gives d=1/4. So the common difference of the given sequence is 1/4.
     
  2. Given a sequence: 1/3, 1/6, 1/9, …
    The common difference of the given sequence is d= 1/9 - 1/6 gives d= -1/18. So the common difference of the given sequence is -1/18.
     

General Term

As we have learned earlier, the general term of an arithmetic progression is represented as a, a+d, a+2d, a+3d, … The corresponding Harmonic Progression of the given A.P. will be 1/a, 1/(a+d), 1/(a+2d),..., where 1/a represents the first term, and the common difference is represented by ‘d’. The general term or the nth term of a harmonic progression is the reciprocal of the nth term of an A.P. can be represented as

an = 1/[a + (n-1)*d]

Here, a represents the first term of an A.P.,
n is the number of terms in the A.P.,
d represents the common difference.

Examples

  1. Determine the 5th and 6th terms of the H.P. 6, 4, 3, …
    The A.P. of the given H.P. is 1/6, 1/4, 1/3, …
    The common difference of the A.P. will be 1/4 - 1/6 = 1/12.
    The nth term of an A.P. can be calculated using an= a+ (n-1)*d
    5th term = 1/6 + 4*(1/12)
    5th term = 1/2
    Similarly, the 6th term of the A.P. = 1/6 + 5*(1/12)
    6th term = 7/12
    Hence, the 5th and 6th terms of the H.P. will be 2 and 12/7, respectively.
     
  2. Find the 50th of an H.P. if the 10th and 20th terms are 20 and 40, respectively.
    If the 10th and 20th terms are 20 and 40, then the 10th and 20th terms of the corresponding A.P. will be
    10th => a + 9d = 1/10 — (1) 
    20th=> a + 19d = 1/20 — (2)

    Solving equations (1) and (2), we will get a= 29/400 and d=-1/400.
    Now, to find the 50th of an A.P. we will write a+ 49d => 29/400 + 49*(-1/400)
    Solving this, we will get the 50th as -1/20.
    Hence, the 50th term of the H.P. will be -20.
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Sum of n terms in an H.P.

The sum of the first n terms of a harmonic progression is given by

Sn= (1/d)ln[(2a+ (2n-1)*d)/(2a-d)] 

Here, Sn represents the sum of first n terms.

ln represents the natural logarithm.
n is the number of terms in the sequence,
a is the first term of the sequence,
d is the common difference of the sequence

Examples

  1. If the sum of the reciprocals of the first 11 terms is 110, find the 6th term.
    The reciprocal of the first 11 terms will be an A.P.
    Sn = n/2[2a + (n-1)d]
    110 = 11/2[2a + (10)d]
    10= a+5d
    Hence, the 6th term of the A.P. is 10. Therefore, the 6th term of the H.P. will be 1/10.
     
  2. The H.P.'s 2nd and 5th terms are 3/4 and 1, respectively. Compute the sum of the 6th and 7th terms of the H.P.
    The 2nd and 5th terms of the corresponding A.P. will be 4/3 and 1, respectively.
    Mathematically it can be written as       a+d = 4/3 — (1)
                                                                   a+4d = 1 —- (2)

    Solving equation (1) and (2) we will get a= 13/9 and d= -1/9.
    Hence, the 6th and 7th terms of the A.P. will be a+5d = 8/9 and a+6d = 7/9.
    Therefore, the 6th and 7th terms of the H.P. will be 9/8 and 9/7.
    a6 + a7 = 9/8 + 9/7
    => 135/56.
     

Applications of Harmonic Progression

  1. Scientists use it to conclude the value of their experiments.
  2. It is used to establish how water boils each time the temperature changes with the same value.
  3. The notes in music use the concept of harmonic sequence.
  4. It is used to calculate the amount of rainfall.
  5. It can be used to calculate the focal length of a lens.

FAQ

  1. Find the 16th term of an H.P. if the 5th and 9th terms are 9 and 18, respectively.
    The 5th and 9th terms of the corresponding A.P. will be 1/9 and 1/18, respectively.
    a+4d = 1/9   —- (1)
    a+8d = 1/18 —- (2)

    Solving equations (1) and (2) we will get a= 1/6 and d= -1/72.
    Hence, the 16th term will be a+15d => (1/6) + 15*(-1/72)
    16th term = -1/24
    Therefore, the 16th term of the H.P. is -24.
     
  2. Find the 4th and 8th terms of the series 6, 4, 3…
    The reciprocals of the given series form an A.P. (1/6, 1/4, 1/3, …) with a common difference of 1/12.
    Hence, the 4th term = a+3d and 8th term = a+7d.
    a4 = 1/6+ 3(1/12) => a4 = 5/12
    a8 = 1/6 + 7(1/12) => a= 3/4 

    Therefore, the 4th and 8th terms of the given series will be 12/5 and 4/3, respectively. 

Key Takeaways

In this article, we have extensively discussed Harmonic Progression. A Harmonic Progression is defined as a sequence of real numbers determined by taking the reciprocal of an arithmetic sequence with non-zero elements. 


Check out this problem - Subarray Sum Divisible By K

Recommended Readings:

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