**Harmonic Progression (H.P.)**

**Definition:**

· A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.

Standard form: 1/a, 1/(a +d), 1/(a +2d)............ 1, 1/2, 1/3,..............are in harmonic progression and 1, 2, 3.....are in arithmetic progression.

**General term of an H.P.**

**Harmonic Mean:**

· If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.

Also, if *a*, *H*, *b* are in H.P., then *H* is called harmonic mean between *a* and *b*.

**Insertion of Harmonic Means: **

Single H.M. between *a* and *b = 2ab/(a +b)*

*H*, H.M. of *n* non-zero numbers a_{1}, a_{2}, a_{3}, a_{4},......a_{n} is given by ;

Let *a*, *b* be two given numbers. If *n* numbers H_{1}, H_{2}, H_{3}.....H_{n} are inserted between *a* and *b* such that the sequence a H_{1}, H_{2}, H_{3}.....H_{n} b is a H.P., then H_{1}, H_{2}, H_{3}.....H_{n }are called *n* harmonic means between *a* and *b*.

Now, a H_{1}, H_{2}, H_{3}.....H_{n} b are in H.P.

1/a, 1/H_{1}, 1.H_{2}, 1/H_{3}.....1/H_{n,} 1/b are in AP.

Let *D* be the common difference of this A.P. Then,

Thus, if *n* harmonic means are inserted between two given numbers *a* and *b*, then the common

difference of the corresponding A.P. is given by

**Properties of H.P. :**

No terms of HP can be zero

** **If *H* is the H.M. between *a* and *b*, then