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:
 Arithmetic progressions
 Geometric progressions

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 nonzero 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.
 1/4, 1/7, 1/10, 1/13, ...
 1, 1/3, 1/5, 1/7, …
 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_{n1.}
Consider the following examples:

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.

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 + (n1)*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

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+ (n1)*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.

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.