Harmonic Progression Examples

The harmonic progression or the harmonic sequence is acquired by picking the reciprocal of arithmetic progression terms. These terms are present in the following form; 1/a, 1/( a + d ), 1/( a + 2d ), 1/( a + 3d ), 1/( a + 4d ), 1/( a + 5d )…1/(a + ( n – 1 )d). Like an AP, i.e., Arithmetic Progression, the nth term can also be computed; it is the total of the n terms belonging to the harmonic sequence. Harmonic progressions and the harmonic mean concepts are widely applicable in mathematics, business, physics and various other fields. Moreover, it has multiple formulas and examples.

Definition

The harmonic progression can be made by the process of taking the reciprocal of the AP, i.e., arithmetic progression’s terms. In case, the AP’s terms provided are; a, a + d, a + 2d, a + 3d, a + 4d,… then the harmonic sequence or harmonic progression’s terms will be; 1/a, 1/( a + d ), 1/( a + 2d ), 1/( a + 3d ), 1/( a + 4d ), 1/( a + 5d ),… Here, ‘a’ is the initial term and ‘d’ becomes the common difference. Both ‘a’ and ‘d’ contain non-zero values. Notably, the series of harmonic progressions has an infinite nature.

Leaning Tower of Lire

The leaning tower of lire is a perfect example of a harmonic progression. A cluster of cubes having identical sides is placed over each other to get as much sideways as possible or sideways distance. The placement of the blocks is done in 1-way distances; 1/2, 1/4, 1/6, 1/8, 1/16,… one-on-one over each other. Furthermore, the motive of the stacked arrangement given above is to get the utmost sideway distances so that the centre of gravity becomes well maintained and the stack doesn’t collapse.

Formulas of Harmonic Progression

Following are the formulas that are useful for various calculations concerning harmonic progression:

Harmonic Progression’s nth term

It is said to be the reciprocal of the AP’s nth term. Harmonic progression’s  nth  term is basically the overall summation reciprocal that belongs to the first term, including the ‘n – 1’ times of the ‘d,’ i.e., the common difference. Moreover, the nth term is helpful in finding out any type of terms belonging to the harmonic progression.

The nth  term of a Harmonic progression or Harmonic Sequence = 1/(a + ( n – 1 )d)

Harmonic Mean

While talking about a harmonic sequence or harmonic progression, whichever term you want in the series can be considered its adjacent term’s harmonic mean.

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

The Harmonic Mean for 2 terms ‘a’ and ‘b’ = (2ab)/( a + b )

The Harmonic Mean for 3 terms ‘a’, ‘b’, and ‘c’ = (3abc)/( ab  +  bc  +  ca )

·   Total summation of the n terms that belong to a harmonic sequence or progression = 1d.log( 2a + ( 2n – 1 )d2a – d )1d.log( 2a + ( 2n – 1 )d2a – d )

Relationship in AM, GM, & HM

Here, we have the relationship among AM, GM and HM with respect to the provided set that belongs to the arithmetic mean (AM), the geometric mean (GM) and the harmonic mean (HM). The arithmetic mean becomes superior, continued by the geometric mean and harmonic mean. Here, ‘AM’ is greater than ‘GM’ and  ‘GM’ is greater than ‘HM.’

The geometric mean’s square is equivalent to the result of the multiplication of the harmonic mean and the arithmetic mean:

GM2 = HM x AM

Harmonic Sequence Applications

The harmonic sequence, along with the harmonic mean, comprises various applications in different areas of mathematics, physics, engineering and business. Here are a few of the essential applications that belong to the harmonic series:

●  The standard average speed of a motor vehicle among 2 sets of equivalent distances is computable with the help of the particular speed’s harmonic mean. In case, the vehicle has a speed of ‘x’ metres/hour for the initial ‘d’ miles and then it is ‘y’ metres/hour for the upcoming ‘d’ miles, then through the entire distance, the vehicle will have an average speed that will be equivalent to the harmonic mean for the 2 given speeds. So, the average speed here will be = ( 2xy )/( x + y ).

●  The concentration of a substance’s mix or the alloy’s density, which comprises two or more than two substances having equal weight and equivalent weight proportion composition is computable with the help of the harmonic mean that belongs to the individual component’s densities.

●  A lens’s focal length is equivalent as compared to the harmonic mean given for the distance that belongs to the object ‘u’ from the lengths and the distance for the given image ‘v’ from the lens; (1/f)=( 1/u + 1/v ).

●  Geometrically, the radius that belongs to the triangle’s incircle is equivalent to 1/3rd of the harmonic mean that belongs to the triangle’s altitudes.

●  In the area of finance, the ratio of the profit earnings can be computed with the help of the weighted-harmonic-mean’s concept for the individual components.

Conclusion

The notes on harmonic progression examples conclude that the harmonic progression or the harmonic sequence is made with the process of taking the reciprocals of an arithmetic progression. Moreover, the concepts of harmonic progression are broadly accepted in areas like mathematics, business, physics and multiple other fields as well at the same time. Furthermore, there are numerous harmonic progression examples and formulas that are useful in all the fields and areas mentioned above. Notably, the harmonic progression examples’ importance is also there in these areas.