In this article we learn about *harmonic progression formula* *for nth term, sum of terms, properties with examples.*

*Harmonic mean formula* for two quantities, three quantities and number of quantities.

## Harmonic Progression and Harmonic Mean formulas with properties

**Definition of Harmonic Progression
**

A series of terms is known as a **Harmonic progression** series when the reciprocals of elements are in *arithmetic progression.*

In general, If** p, q, r, s** are in arithmetic progression then** 1/p, 1/q, 1/r, 1/s** are all in Harmonic progression.

**For example** take the series 1/10, 1/12, 1/14, 1/16 . . . . . . . .

Here reciprocal of the given series is 10, 12, 14, 16 . . . . . . .

The series is clearly in Arithmetic progression having first term is ’10’ and common difference is ‘2’

So the given series is in Harmonic progression

**Harmonic progression nth term & sum formula**

In order find the nth term or sum of terms in a Harmonic Progression, one should make the series into corresponding arithmetic series and then find nth term of the series.

Let 1/a, 1/(a+d), 1/(a + 2d), . . . . . . is in an HP then the inverse of a harmonic progression follows the rule of an arithmetic progression. So

a, a + d, a + 2d, . . . . . . . are in AP and first term is ‘a’ , common difference is “d”

nth term of A.P **a _{n} = a + (n -1) d**

So nth term of given HP is**
**

Similarly sum of ‘n’ terms of AP is S_{n}

Sum of ‘n’ terms of given HP is

**Note: **The sum of a harmonic series will never be an integer except first term is “1” and number of terms are “1”.

**Properties of Harmonic Progression**

**1.** Any number of quantities are said to be in harmonic progression when every three consecutive terms are in harmonic progression.

**2.** Three quantities p, q , r are said to be Harmonic Progression

So 1/p, 1/q , 1/r are in arithmetic progression

Common difference of the series d = 1/q – 1/p = 1/r – 1/q

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**3.** Selection of terms in Harmonic Progression

Number of terms | Terms |

3 | 1/(a–d), 1/a, 1/(a+d) |

4 | 1/(a–3d), 1/(a–d), 1/(a+d), 1/(a+3d) |

5 | 1/(a–2d), 1/(a–d), 1/a, 1/(a+d), 1/(a+2d) |

### Harmonic mean between given quantities

**Harmonic mean between two quantities**

Let **p, q** be the two quantities and H is a harmonic mean of their quantities. Then **1/p, 1/H and 1/q** are in Arithmetic progression

Therefore

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i.e if **p, q & r** be the three quantities are in* harmonic progression* then is a *harmonic mean* of their quantities.

Therefore, harmonic mean formula

Harmonic mean

**Harmonic mean between number of quantities**

If r_{1}, r_{2}, r_{3}, r_{4}, . . . . . r_{n} be an finite “n” non-zero numbers are in HP. Then harmonic mean “H” of these numbers is given by

1/H = (1/n) [ 1/ r_{1 }+ 1/r_{2 }+ 1/r_{3 }+ 1/r_{4},+. . . . . +1/ r_{n }]

**Related Articles**

Arithmetic Progression Formulas

Arithmetic Progression Questions with Solutions

Arithmetic Mean formula with Examples

Geometric Progression formulas

Geometric Progression Examples

Harmonic Progression formulas

Relation between AM, GM and HM