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Harmonic Progression (HP)

A sequence a1, a2, a3, …, an, …. of non-zero numbers is called a harmonic progression or a harmonic sequence, if the sequence 63114.png, …, 63108.png, … is an arithmetic progression.
 
nth term of an HP The nth term of an HP is the reciprocal of the nth term of the corresponding AP. Thus, if a1, a2, a3, …, an, is an HP and the common difference of the corresponding AP is d, i.e., d = 63102.png, then the nth term of the HP is given by an = 63096.png.
 
In other words, nth term of an HP is the reciprocal of the nth term of the corresponding AP.

Insertion of Harmonic Means

Let a, b be two given non-zero numbers. If n numbers H1, H2, …, Hn are inserted between a and b such that the sequence a, H1, H2, H3, …, Hn, b is an HP, then H1, H2, …, Hn are called n harmonic means between a and b.
 
Now, a, H1, H2, …, Hn, b are in HP.
 
63090.png, 63084.png are in AP
 
Harmonic means of two given numbers a and b is H = 63078.png.

AM, GM, and HM of two positive real numbers

Let A, G, and H be arithmetic, geometric, and harmonic means of two positive numbers a and b, Then,
A = 63072.png, G = 63066.png and H = 63060.png
 
These three means possess the following properties:
  1. AGH.
  2. A, G, H form a GP, i.e., G2 = AH.
  3. If A and G be the AM and GM between two positive numbers, then the numbers are A ± 63054.png.
  4. The equation having a and b as its roots in x2 – 2Ax + G2 = 0.
  5. If A, G, H are arithmetic, geometric, and harmonic means of three given numbers a, b, and c, then the equation having a, b, c as its roots is x3 – 3Ax2 + 63047.png xG3 = 0.




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