Harmonic Progression

In Mathematical terms, a progression is basically a number series which follows a specific pattern. Building on this definition, a Harmonic Progression (HP) can be defined as a series of real numbers which is calculated by taking reciprocals of the arithmetic progression reciprocals which do not contain 0. Any term in this type of sequence is regarded as the harmonic means of its two neighbours. 

For example, the series a, b, c, d, for example, is called an arithmetic progression, the harmonic progression may be written as 1/a, 1/b, 1/c, 1/d.  

What is Harmonic Mean?

The harmonic mean of a series is the reciprocal of the arithmetic mean of the reciprocal values in the series. 

Harmonic Mean = n / [(1/a) + (1/b) + (1/c) + (1/d)] 

Harmonic Progressions Formula

The term at the nth place of a harmonic progression is the reciprocal of the nth term in the corresponding arithmetic progression. This can be mathematically represented by the following formula.

nth term of H.P = 1/[a + (n-1)d]

a — First Term in A.P

d— Common Difference

n — Number of Terms in A.P 

Sum of Harmonic Progression Formula

Let's consider 1/a, 1/a + d, 1/a + 2d, 1/a + (n-1)d as a given harmonic progression. Now, to calculate the sum of every single element in this progression i.e. the sum of the harmonic progression, we use the following formula.

Sn = (1/d) x ln [{2a + (2n−1)d} / (2a−d)]

Comparison: Arithmetic Progression, Geometric Progression, and Harmonic Progression

Relationship Between AP, GP, and HP

For any two numbers, if A, G, H are respectively the Arithmetic, Geometric, and Harmonic Mean, then the relationship between those three is given by the following formula. 

G.M2 = A.M x H.M, where A.M., G.M., H.M are in geometric progression A.M ≥ G.M ≥ H.M 

Solved Harmonic Progression Problems

Example 1: Determine the 5th term and the 49th term of the harmonic progression 6, 4, 3,…

Solution:

H.P = 6, 4, 3

The arithmetic progression for the given H.P is A.P = ⅙, ¼, ⅓, ….

Here T2-T1 = T3-T2 = 1/12, so, 1/12 is the common difference. 

d=1/12

So, in order to find the 5th term of the A.P, use the formula:

The nth term of an A.P = a + (n-1)d

Here, a = ⅙, d= 1/12

Now, we have to find the 5th term,

So, take n=5

Now put the values in the formula, we have

5th term of the A.P = (⅙) + (5-1)(1/12)

= (⅙) + (4/12)

= (⅙) + (1/3)

= 3/6= 1/2

Therefore, the fifth term of the arithmetic progression is 1/2. The nth term of a harmonic progression is the reciprocal of the nth term in the corresponding arithmetic progression.

Therefore, the fifth term of the harmonic progression is the reciprocal of 1/2, which is equal to 2. 

In order to find the 49th term of the A. P, use the formula:

The nth term of an A.P = a + (n-1)d

Here, a = ⅙, d= 1/12

Now, we have to find the 50th term,

So, take n=49

Now put the values in the formula, we have

50th term of the A.P = (⅙) + (49-1)(1/12)

= (⅙) + (48/12)

= (⅙) + (4)

= 25/6

Therefore, the forty-ninth term of the arithmetic progression is 25/6. Hence, the forty-ninth term of the harmonic progression is the reciprocal of 25/6, which is equal to 6/25, which is equal to 0.24. 


Example 2: Compute the 100th term of HP if the 10th and 20th term of HP are 20 and 40 respectively.

Solution:

The corresponding A.P to the given H.P is given below:

10th Term of A.P = a + 9d = 1/20 —(1)

20th Term of A.P = a + 19d = 1/40 —(2)

By solving these two equations, we get

a =29/400 and d = -1/ 400

To find 100th term, we should write the expression in the form,

a + 99d = (29/400) + 99(-1/400)

= (29/400) - (99/400)

= (-70/400)

=(-7/40)

Thus, the 100th term of the H.P = 1/(100th term of the A.P) = (-40/7)

Therefore, the 100th term of the H.P is (-40/7).

FAQ (Frequently Asked Questions)

1. How Do We Write an Arithmetic Progression From a Harmonic Progression?

A Harmonic Progression (HP) is defined as a series of real numbers which is calculated by taking reciprocals of the arithmetic progression reciprocals which do not contain 0. The term at the nth place of a harmonic progression is the reciprocal of the nth term in the corresponding arithmetic progression.

So, if you are given an arithmetic progression that goes a, b , c , d. You can write the corresponding harmonic progression as 1/a, 1/b, 1/c, 1/d. 

2. What is an Application of Harmonic Progressions? 

The Leaning Tower of Lire is an outstanding example of a Harmonic Progression. In it, rectangular blocks are placed on top of each-other to reach the maximum lateral distance or sideways lean. The bricks are laterally stacked 1/2, 1/4,1/6, 1/8, 1/10 units below the initial pin. This ensures that the centre of gravity is at the centre of the structure just enough, to prevent it from collapsing. Even a slight increase in weight on this delicate framework can render it unstable and cause its collapse.