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 an = a + (n -1) d
So nth term of given HP is
Similarly sum of ‘n’ terms of AP is Sn
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
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
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 between number of quantities
If r1, r2, r3, r4, . . . . . rn 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/ r1 + 1/r2 + 1/r3 + 1/r4,+. . . . . +1/ rn ]
Harmonic Progression formulas