Harmonic Progression (HP)

A sequence a1, a2, a3, â€¦, an, â€¦. of non-zero numbers is called a harmonic progression or a harmonic sequence, if the sequence , â€¦, , â€¦ 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, â€¦, an, is an HP and the common difference of the corresponding AP is d, i.e., d = , then the nth term of the HP is given by an = .

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, â€¦, Hn are inserted between a and b such that the sequence a, H1, H2, H3, â€¦, Hn, b is an HP, then H1, H2, â€¦, Hn are called n harmonic means between a and b.

Now, a, H1, H2, â€¦, Hn, b are in HP.

â‡’ , are in AP

Harmonic means of two given numbers a and b is H = .

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 = , G = and H =

These three means possess the following properties:
1. A â‰¥ G â‰¥ H.
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 Â± .
4. The equation having a and b as its roots in x2 â€“ 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 â€“ 3Ax2 + x â€“ G3 = 0.