Harmonic Progression

Harmonic progression can be stated as a sequence or progression that is used to represent reciprocal arithmetic progression. As all progression follows a systematic pattern, harmonic progression also follows a set of rules and is written in a predictable pattern. This article revolves around harmonic progression, its general term and sum of a harmonic progression. A formula is also provided in this article that is based on how to find harmonic progression. 

Mathematical explanation of harmonic progression

Harmonic progression is a type of sequence that represents arithmetic progression in reciprocal form. It is found that each term of a harmonic progression is a harmonic mean of its neighbouring terms. Arithmetic progression used for making harmonic progression must not contain zero as its reciprocal is not defined yet. 

Application of harmonic progression

Harmonic progression is widely used to solve sums of mathematics. Apart from that, it is also used to make reciprocal arithmetic progression. Other than mathematics, it is also applied in geometry to find out about collinear points. In geometry, harmonic progression is applied to find outsides of a triangle if its altitude is following arithmetic progression. 

General form of harmonic progression

General term of harmonic progression can be represented as a reciprocal of progression that is in arithmetic form. For representation of harmonic progression, an arithmetic progression is required and then after its reciprocal is done to convert it into harmonic progression. 

Suppose a general form of an arithmetic progression is represented as a, b, c, d, e, f . . . then harmonic progression can be represented as 1/a, 1/b, 1/c, 1/d, 1/e, 1/f . . . . and so on. 

Similarly, if terms of a sequence following arithmetic progression is given as “a, a + d, a + 2d, a + 3d, a + 4d,” . . . . And so on where “a” is first term and d is a common difference of arithmetic progression. 

Then harmonic progression can be formulated as ” 1 / a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a + 4d), . . . . . .” and so on.

General formula for harmonic progression

General formula for harmonic progression can be stated as;

1 / h, 1 / (h + d), 1 / (h + 2d), 1 / (h+ 3d), 1 / (h + 4d), . . . . . . . 1 / (h + nd)

Where, 1 / h is the first term and d is the common difference of initial arithmetic progression; and 1 / (h + nd) is the nth or last term of harmonic progression. 

How to find harmonic progression of nth term

A general rule is needed to find specific terms of a sequence. In this regard, this section is based on how to find harmonic progression. The nth term of a harmonic progression can be represented in form of formulae:

nth term of a H.P = 1 / [a + (n – 1)d]

Or, nth term of an H.P. = 1 / (nth term of initial arithmetic progression)

In this equation, a = first term of initial arithmetic progression,

d = common difference, and

n = number of terms of initial arithmetic progression. 

Properties of harmonic progression

Every progression follows a set of rules that can be stated as its properties. Each sequence of a progression must follow these properties. Some of properties are mentioned below:

• If every three consecutive terms of any number series are in harmonic progression then whole sequence is said to be in harmonic progression. 
• Three numbers a, b, c are said to be in harmonic progression only if 1/a, 1/b, 1/c must be in arithmetic progression. 

Sum of harmonic progression

To find the sum of a harmonic progression, it must require a sequence in harmonic progression. In this particularity, 1/a, 1/(a+d), 1/(a+2d), . . . . , 1/[a+(n-1)d] is taken as a sequence following harmonic progression. Then, formula used for formula string sum of finite terms of harmonic progression can be given as:

Sum of n terms of H.P.; Sn = (1/d)ln {[2a + (2n – 1)d] / (2a – d)}

To justify above equation, Sn = sum of harmonic progression having n number of terms,

ln = log or natural logarithm,

a = first term of corresponding arithmetic progression, and 

d = common difference of corresponding arithmetic progression.

Conclusion

Harmonic progression is found to be calculated under arithmetic progression. If a sequence of arithmetic progression is given then it will be easy to derive harmonic progression. Mathematical explanation about harmonic progression is given in this article. General form and formula are also discussed to justify harmonic progression with help of mathematical formulae. Two properties are also elaborated in this article that will provide general characteristics of a sequence in harmonic progression. It is difficult to find some of a sequence in a haphazard manner which is why the formula for finding sum of harmonic progression is also discussed.