Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM)
To know the relation between the AM, GM and HM, we need the formulas of all these three types of mean.
Suppose that “r” and “s” are the two numbers and the number of values = 2, then
AM = (r+s)/2
⇒ 1/AM = 2/(r+s) ……. (1)
GM = √(rs)
Taking square both side
⇒GM2 = rs ……. (2)
HM= 2/[(1/r) + (1/s)]
⇒ HM = 2/[(r+s)/rs]
⇒ HM = 2rs/(r+s) ….. (3)
Now, put (1) and (2) in (3), we get
HM = GM2 /AM
⇒GM2 = AM × HM
GM = √[ AM × HM]
Hence, the relation between AM, GM and HM is
GM2 = AM × HM
Therefore the square of the Geometric Mean is equal to the product of the Arithmetic Mean and the Harmonic Mean.
Harmonic Progression
A Harmonic Progression (H.P.) is a mathematical sequence generated by taking the reciprocals of an Arithmetic Progression. In this sequence, each term is the harmonic mean of its adjacent terms, this series is called Harmonic Progression.
A Harmonic Progression of separate unit fractions cannot add to an integer (unless in the specific case where (a = 1 and d = 0). The reason lies in the fact that the progression will contain at least one denominator divisible by a prime number that does not a divisor of any other denominator. Harmonic Progression is also called Harmonic Sequence.
In this article, we will discuss the definition, applications, and formula of Harmonic Progression, and understand the difference and relation between arithmetic mean, geometric mean, and harmonic mean to calculate Harmonic Progression in mathematics.
Table of Content
- What is Harmonic Progression (HP)?
- Harmonic Progression Example
- Harmonic Progression Formula
- Harmonic Progression Formula for nth Term
- Harmonic Progression Sum
- What is Harmonic Sequence?
- Harmonic Mean
- Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM)
- Applications of Harmonic Progression
- Solved Examples on Harmonic Progression
- Harmonic Progression Questions