# Formulas For Harmonic Progression

## Formulas For Solving Harmonic Progression Easily

**Harmonic Progression is a sequence of quantities whose reciprocals are in arithmetical progression .Each term in the Harmonic Progression is the Harmonic Mean Of its Neighbouring Term. Hence , it is is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. mean of its two neighbors.**

**Definition & Formula for Harmonic Progression**

- Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression
- Also, a sequence is a harmonic progression when each term is the harmonic mean of its neighbouring termTo solve a problem on Harmonic Progression, make the corresponding AP series and then solve the problem. For Harmonic Progression sum of integer is not possible.
- For example: Given sequence or series is 1/2, 1/4, 1/6, 1/8, …. Take the reciprocal of each term, therefore, we get 2, 4, 6, 8,…… Now, a = 2 and d = 2

**Formulas of Harmonic Progression (HP)**

**How to find n**^{th}term of an HP

_{Tn} = 1/(a + (n – 1)d)

where t_{n} = nth term,

a= the first term ,

d= common difference,

n = number of terms in the sequence

**Harmonic Mean (HM)**

Harmonic Mean is type of numerical average, which is calculated by dividing the number of observation by the reciprocal of each number in series.

If a, b are in HP, then there HM is

**HM = n / { 1/a _{1 }+ 1/a_{2 }… 1/a_{n}}**

Where, n = Total number of numbers or terms, a_{1}, a_{2},…..a_{n } = Individual terms or individual values.

**Properties of Harmonic Progression**

- For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then
**A≥ G ≥ H** - Relationship between arithmetic, geometric, and harmonic means
**AM * HM = GM**^{2}

- Unless a = 1 and n = 1, the sum of a harmonic series will never be an integer. This is because at least one denominator of the progression is divisible by a prime number that does not divide any other denominator.
- Three consecutive numbers of a HP are: 1/(a–d), 1/a, 1/(a+d)
- Four consecutive numbers of a HP are: 1/(a–3d), 1/(a–d), 1/(a+d), 1/(a+3d).

**Read Also – How to solve Harmonic Progression Quickly**

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