Harmonic Progression Introduction
In Mathematical terms, a progression is a number series that 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 neighbors.
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 does Harmonic Mean?
The harmonic mean of a series is the reciprocal of the arithmetic mean of the reciprocal values in the series.
Harmonic Mean = \[ \frac{n}{(1/a)+(1/b)+(1/c)+(1/d)} \]
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.
The nth term of \[H.P=\frac{1}{a+(n-1)d}\]
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
\[\frac{2a+(2n-1)d}{(2a-d)}\]
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, 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.
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 terms 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 the 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).
FAQs on Harmonic Progression
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.
3. What is a Harmonic Progression?
A set of numbers is said to be Harmonic Progression if the reciprocal of the series are in Arithmetic Progression. For example, a,b,c,d,.... Are said to be in Harmonic Progression if 1/a,1/b,1/c,1/d are in Arithmetic Progression. If a, b are in Harmonic Progression, then the nth term of the progression could be calculated by the formula
The nth term of H.P = 1/(nth term of AP). Here, H.P stands for Harmonic Progression and A.P stands for Arithmetic Progression. HM is always less than the geometric and the Arithmetic Progressions of the series. The best example of a series of Harmonic Progression is ½,¼, ⅙, ……, as the reciprocals i.e, 2,4,6,8 are in Arithmetic Progression. To solve any problem related to the Harmonic Progressions, we should use Arithmetic Progression.
4. What is the difference between Harmonic Progression and Arithmetic Progression?
A series of numbers are said to be in Arithmetic Progression if the difference between two consecutive digits in the series is a constant value. For example, 2,4,6,8 are said to be in Arithmetic Progression as the difference between two consecutive numbers is “2”. On the other hand, a series of numbers is said to be in Harmonic Progression if their reciprocals are said to be in Arithmetic Progression. For example, a,b,c,d…are said to be in Harmonic Progression if 1/a,1/b,1/c,1/d …are in Arithmetic Progression. So the difference between two consecutive terms like 1/a,1/b should be “k” and k is constant. The relation between the harmonic and Arithmetic Progression is, the Harmonic Progression for any two numbers is always less than the Arithmetic Progression of the same two terms.
5. What is the difference between Harmonic and Geometric Progression?
A series of numbers are said to be in geometric progression when the succeeding term of a number in the series is produced by multiplying the number with a certain constant value. For example, 3,6,12,24…. Are in geometric progression as the succeeding term is defined by multiplying the preceding term with constant value “k” in this case, “2”. On the other hand, Harmonic Progression is where the set of number’s reciprocals is in Arithmetic Progression. For example, x,c,v,b…. Are in Harmonic Progression if the reciprocals: 1/x,1/c,1/v,1/b are in Arithmetic Progression which means the difference between two consecutive terms should be constant throughout the series.
The relation between geometric and Harmonic Progression is such that the geometric progression of two numbers is always higher than the Harmonic Progression.
6. What are the real-life examples and applications of Harmonic Progression?
Harmonic Progression is when the reciprocals of a series of numbers are in Arithmetic Progression which means the difference between two consecutive numbers is constant throughout the series. Following are the real-life examples where the Harmonic Progression is used:
Used by the scientists in conducting their experiments
Used to determine how water boils every time the temperature changes with a constant value
Notes of a music tape use the Harmonic Progression to sequence the melody
It could be used to measure the number of raindrops when rainfall takes place in a place and the raindrops are infinite.
A very close real-life example of Harmonic Progression is traffic bunches.
To know more about the Harmonic Progression, formulas related to it, examples of problems with solutions, please read the above article in Vedantu.
7. What is the role played by progressions in explaining the Malthusian theory?
Malthus explains his work about the population using the concept of progressions in his work named “Principle of Population”, he says that the population grows exponentially in geometric progression and the food, other resources grow slowly in Arithmetic Progression. In geometric progression, the successive number is produced by multiplying the previous number with a constant value throughout the series. In Arithmetic Progression, the difference between two consecutive numbers is constant. Malthus says that the population grows in geometric progression and the resources grow in Arithmetic Progression. This will put a strain on the resources available and conflicts arise out of such constraints. This theory also explains how one utilizes limited resources optimally to meet the unlimited needs of human beings. It also reminds us of the need for sustainable utilization of resources.