Harmonic Progression

Harmonic progression is obtained by taking the reciprocal of an arithmetic progression’s terms. A harmonic progression has the following terms: 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),… 1/(a + (n – 1)d). We can compute the nth term, the sum of n harmonic progression terms, similarly to the arithmetic progression.

A Harmonic Progression (HP) is a sequence of real numbers formed by calculating the reciprocals of the arithmetic progression that does not contain 0. Any phrase in the sequence is considered the harmonic mean of its two neighbours in harmonic progression. The sequence a, b, c, d,… is an example of an arithmetic progression; the harmonic progression can be represented as 1/a, 1/b, 1/c, 1/d,…

Harmonic Mean: The reciprocal of the arithmetic mean of the reciprocals is used to determine the harmonic mean. The harmonic mean can be calculated using the following formula: 

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

Where the values are a, b, c, and d, and n is the number of values present.

Harmonic Progression Formula

We must find the corresponding arithmetic progression sum to answer the harmonic progression difficulties. It signifies that the nth term of the harmonic progression is equal to the reciprocal of the corresponding A.P.’s nth term. As a result, the formula for determining the nth term of the harmonic progression series is as follows:

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

Where “a” is A.P’s initial word.

“d” represents the common difference, and “n” represents the number of phrases in A.P.

The preceding formula can also be written as:

The nth term of H.P = 1/ (nth  term of the corresponding A.P)

Harmonic Progression Sum

If a harmonic progression of 1/a, 1/a+d, 1/a+2d,…., 1/a+(n-1)d is given, the formula to compute the sum of n terms in the harmonic progression is given by the formula:

The sum of n terms,

Where “a” is A.P’s first term, “d” is A.P’s common difference, and “ln” is the natural logarithm.
Application of Harmonic Sequence

Harmonic sequence and harmonic mean have a wide range of applications in mathematics, engineering, physics, and business. Some of the most important uses of harmonic series are as follows.

• The harmonic mean of the various speeds can be used to compute the average speed of a vehicle over two sets of equal distances. If the vehicle’s speed is x mph for the first d miles and y mph for the next d miles, the vehicle’s average speed for the entire trip is equal to the harmonic mean of these two speeds. (2xy) / (x + y) = Average Speed

• Using the harmonic mean of the densities of the individual components, the density of a mixture or the density of an alloy of two or more substances of identical weight and percentage composition can be determined.

• The focal length of a lens is equal to the harmonic mean of the object’s (u) distance from the lengths and the image’s (v) distance from the lens. 1/f equals 1/u + 1/v.

• In geometry, the radius of a triangle’s incircle equals one-third of the harmonic mean of the triangle’s altitudes.

• In finance, the profit-earnings ratio is calculated using the weighted harmonic mean of each component.

Conclusion

The harmonic progression is constructed by taking the reciprocal of the arithmetic progression’s terms. If the arithmetic progression terms are a, a + d, a + 2d, a + 3d,…., then the harmonic progression (or harmonic sequence) terms are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),…… The first term is a, and the common difference is d. a and d both have non-zero values.