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A harmonic progression is a mathematical progression that is created by taking the reciprocals of an arithmetic progression. When each term is the harmonic mean of the neighbouring terms, a sequence is equivalently a harmonic progression.
The reciprocal of the terms of an arithmetic progression yields a harmonic progression. 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),…1/(a + (n – 1)d are the terms of a harmonic progression. We can compute the nth term, the sum of n terms of the harmonic progression, similarly to the arithmetic progression.
Harmonic Progression Definition
“A harmonic progression (H.P.) is a set of real numbers that do not contain 0 and are derived from the reciprocals of an arithmetic progression.” Any term in a series is considered the harmonic mean of its two neighbours in harmonic progression”.
Formulas for Harmonic Progression
The formulas below are useful for a variety of harmonic progression calculations.
Harmonic Progression nth term
It’s the reciprocal of the arithmetic progression’s nth term. 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 locate any of the harmonic sequence’s terms.
nth=Term of HP= / 1(a+n-1)d
Harmonic mean
To form the arithmetic progression, the harmonic progression is multiplied by its reciprocal. To solve the remaining problems, we must first find the first term and the common difference. The harmonic progression can be solved to determine the nth term or the sum of n terms.
The reciprocal of the terms of the arithmetic progression is used to create the harmonic progression. If the arithmetic progression’s given terms are a, a + d, a + 2d, a + 3d,….,
then the harmonic progressions (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. The values of a and d are both non-zero.
An infinite series of harmonic progressions exist.
Any series term is the harmonic mean of its neighbouring terms in a harmonic progression.
Harmonic Mean = n /[1/a + 1/(a + d)+ 1/(a + 2d) +1/(a + 3d) +…]
Two terms a and b have a harmonic mean =2ab / (a+b)
Harmonic Sequence Sum of n terms
Harmonic sequence = 1d.log (2a+2n-1d)/2a-d
Difference between the Progression and a Sequence
The difference between progression and sequence is shown in the table below.
Progression | Sequence |
Sets of numbers that are arranged according to a specific rule are known as progressions. | A sequence is a group of numbers that are arranged according to a set of rules. |
a progression has a formula for calculating its nth term. | A logical rule, such as a group of prime numbers, can be used to create a sequence. |
Example: A progression is 2, 4, 6, 8, and 10. 2n is the nth term. | Example: A sequence is 2, 3, 5, 7, 11, 13, 17 |
Conclusion
In this article we conclude that“The reciprocals of an arithmetic progression form a harmonic progression, which is a sequence of real numbers. It’s a set of real numbers in which each term is the harmonic mean of its two neighbours.” An infinite series of harmonic progressions exist. The use of harmonic progression to calculate the amount of rainfall creates the illusion that the number of raindrops can be estimated when the series is infinite. Harmonic progression and the harmonic mean have numerous applications in mathematics, physics, business, and other fields.
