The reciprocal of the terms in an arithmetic progression gives us a harmonic progression. Some of the terms of harmonic progression are 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 terms of the harmonic progression like we do with arithmetic progression. Harmonic progression and the harmonic mean are used in mathematics, science, business, and other fields. Let’s look into harmonic progression’s definition and the nth term of harmonic progression. 

What is Harmonic Progression?

The reciprocal of the terms of the arithmetic progression is used to create the harmonic progression. If the arithmetic progression’s terms are a, a + d, a + 2d, a + 3d,……….Then the harmonic progressions (or harmonic sequence) terms will be are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),…… so on. The first term is a, and the common difference given is d. The values of a and d are both non-zero numbers. 

Harmonic progression is equal to infinite series. 

Leaning tower of lire = harmonic progression 

Leaning Tower of Lire: The leaning tower of lire is one good examples of harmonic progression. To obtain maximum sideways or lateral distance, a set of uniform side blocks by stacking one on top of one another. 

The one-way distance is denoted as 1/2, 1/4, 1/6, 1/8, 1/16… have been stacked one over the other. The objective of this stacked structure is to produce lateral distances while maintaining the center of gravity and preventing it from collapsing.

Harmonic Progression Formulas

The formulas given here are used for different harmonic progression computations: 

The nth term of a Harmonic Progression: 

The reciprocal of the nth term of the arithmetic progression is the nth term of a Harmonic Progression. The reciprocal of the sum of the first term and the (n – 1) times of the common difference is the nth term of the harmonic progression. The nth term can be used to point out any of the harmonic sequence’s terms. 

nth Term of HP = 1/(a + (n – 1)d)

Harmonic Mean:

Any phrase in a harmonic progression is the harmonic mean of its neighboring terms.

Harmonic Mean = n /[1/a + 1/(a + d)+ 1/(a + 2d) +1/(a + 3d) +….]

• The harmonic mean of two terms a and b = (2ab) / (a + b).

• The harmonic mean of three terms a, b, and c = (3abc) / (ab + bc + ca)

Sum of n terms of harmonic sequence

• ½ log (2a + (2n -1) d /2a – d) 

Relationship between AM, GM, and HM

For the given set of the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM), the arithmetic mean is greater, followed by the geometric mean, and finally, the harmonic mean. 

AM > GM > HM

The product of the arithmetic and harmonic means equals the square of the geometric mean.

GM2 = AM x HM

Facts about Harmonic Progression

• To solve any problem with Harmonic Progression, you first have to create the AP series and then solve the problem.

• An H.P’s nth term is given by 1/ [a + (n -1) d], just as an A.P’s nth term is given by an = a + (n-1)d.

• If the arithmetic, geometric, and harmonic means of two numbers are A, G, and H, respectively, then. 

• A ≥ G ≥ H

• An H = G², i.e., A, G, H are in GP

• If we need to identify three numbers in an H.P., we can assume they are 1/a–d, 1/a, and 1/a+d.

• Most Harmonic progression questions are answered by first converting them to A.P.

Uses of Harmonic sequence 

Now we know about the nth term of harmonic progression importance, let’s look at harmonic progression uses. In addition to mathematics, engineering, physics, and business, harmonic sequence, and harmonic mean have numerous applications. The following are some of the most important harmonic series applications: 

• The harmonic mean of the various speeds is used to calculate 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 across the entire trip equals the harmonic mean of these two speeds.

Average Speed = (2xy) / (x + y).

• The harmonic mean of the densities of the individual components can be used to calculate the density of a mixture of substances or the density of an alloy of two or more substances of identical weight and percentage composition.

• A lens’ focal length is equal to the harmonic mean of the object’s (u) distance from the lens and the image’s (v) distance from the lens. 

1/f is equal to 1/u Plus 1/v.

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

• The profit-earning ratio is calculated in finance using the concept of the weighted harmonic mean of separate components.

Conclusion 

Progression is a sequence of integers arranged in a predictable manner in mathematics and a specific form of a number, set with well-defined rules. There’s a difference to be made between a progression and a sequence. The nth term of a progression is derived using a specific formula, whereas the nth term of a sequence is calculated using specific logical criteria. This article tells you about the nth term of a harmonic progression and the nth term of harmonic progression examples.