Harmonic Progression

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 n^th 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= a_n - a_n-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 n^th term of a harmonic progression is the reciprocal of the n^th term of an A.P. can be represented as

a_n = 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 5^th and 6^th 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 n^th term of an A.P. can be calculated using a_n= a+ (n-1)*d
   5^th term = 1/6 + 4*(1/12)
   5^th term = 1/2
   Similarly, the 6^th term of the A.P. = 1/6 + 5*(1/12)
   6^th term = 7/12
   Hence, the 5^th and 6^th terms of the H.P. will be 2 and 12/7, respectively.
    
2. Find the 50^th of an H.P. if the 10^th and 20^th terms are 20 and 40, respectively.
   If the 10^th and 20^th terms are 20 and 40, then the 10^th and 20^th terms of the corresponding A.P. will be
   10^th => a + 9d = 1/10 — (1) 
   20^th=> a + 19d = 1/20 — (2)
   
   Solving equations (1) and (2), we will get a= 29/400 and d=-1/400.
   Now, to find the 50^th of an A.P. we will write a+ 49d => 29/400 + 49*(-1/400)
   Solving this, we will get the 50^th as -1/20.
   Hence, the 50^th term of the H.P. will be -20.