Harmonic Progression

A harmonic progression is formed by taking the reciprocals of an arithmetic progression. In arithmetic progression, we add the same number each time, like a (a + d), (a + 2d), (a + 3d),. In a harmonic progression, we take the reciprocal of each term in an arithmetic progression, so we will get 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d).

Here, 
a = first term, 
d = common difference between the terms. 
To make sure the terms are defined and the sequence progresses, so ‘a’ and ‘d’ can never be zero. This pattern continues indefinitely, making it an infinite sequence. 

Let’s understand this clearly using the following example:
Rahul rides a bicycle on a straight road. If he rides at different speeds over different parts of the journey, his travel time follows a harmonic pattern.
If he travels at a speed of 10 km/h, 20 km/h, 30 km/h, 40 km/h,..., these speeds are an arithmetic progression because he adds 10 km/h each time.
But if he calculates the time taken for each part of the journey, the values will be 1/10, 1/20, 1/30, 1/40,..., which forms a harmonic progression.

Harmonic progression is used in average speed calculations, especially when the distances are covered at different speeds and the total time follows a harmonic pattern.
 

The relationship among Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) is: 
AMGMHM

This means that the Arithmetic Mean is always the largest, followed by the Geometric Mean, and the Harmonic Mean is the smallest. 

To understand this, consider any two numbers a and b. The formulas for the arithmetic mean, geometric mean, and harmonic mean are as follows. The arithmetic mean is the average of two numbers.

AM = a+b2

The geometric mean is equal to the square roots of the product of two numbers. 

GM = ab

The harmonic mean of two numbers is equal to the inverse of the average of their reciprocals. 

HM = 2aba+b

The formula for the relation between AM, GM, and HM is the product of the arithmetic and harmonic mean is equal to the square of the geometric mean.

AM × HM = GM2

By deriving this formula, we are able to identify it better.

AM × HM = a+b2 × 2aba+b
    = ab

Here, ab can be derived as ab2 = GM2
 

Harmonic progression has some important formulas that help in calculations. These formulas help us find specific terms, the average value, and the total sum of a harmonic sequence. 

The nth term in HP is the reciprocal of the nth term of an arithmetic progression. The formula for HP is:

Hn = 1a + (n - 1) . d

Where, a is the first term of arithmetic sequence.
d is the difference between the terms.
n is the position of the term to be determined.

Harmonic Mean: It is the type of average used in harmonic progression. It is useful when dealing with speed, distance, or rates. For any two numbers a and b, the harmonic mean is: 
HM = 2aba + b
For any three numbers, the harmonic mean is:

HM = 3abab + bc + ca

Finding the total sum of an HP is more complicated than arithmetic or geometric progressions as it involves fractions. The sum of first n terms is given by:

Sn = 1d log (2a + (2n - 1)d2a - d)

This formula is used in more advanced calculations, especially in science and engineering.
 

Harmonic Progression is an important concept that can be applied to real-life situations where quantities are inversely related. Here are a few real-life applications of Harmonic Progression.

Speed and Travel Time:
When a vehicle moves at different speeds for equal distances, the total time taken follows a harmonic progression. This concept is used in aviation, shipping, and railway scheduling to calculate travel times accurately. For example: The harmonic mean is used to calculate the total time when a car travels 100 km at 50 km/h and another 100 km at 100 km/h. This helps in accurately planning the total travel time.

Electrical Circuits:
When the resistors are connected parallel, the total resistance follows a harmonic progression. Each term is always a reciprocal of the resistance. It also improves the efficiency of electrical circuits, power grids and transformers. 

Finance and Economics:
Harmonic mean is used in finance for averaging values when dealing with ratios, like interest rates, stock price calculations, and investment returns. This is useful in funds, risk analysis, and portfolio management. For example: Suppose that a stock analyst is comparing two stocks with P/E ratios of 10 and 20. The harmonic mean is used in place of the standard average:
HM = (2 × 10 × 20)/ (10 + 20) = 400/30 =13.333