## What is Harmonic Progression in Mathematics?

### Harmonic Progression (H.P.)

**Definition:**

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

### 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.

**(1) Insertion of harmonic means**

(i) Single H.M. between a and b \(=\frac { 2ab }{ a+b }\).

(ii) H, H.M. of n non-zero numbers a_{1}, a_{2}, a_{3}, ….. a_{n} is given by

(iii) 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*.

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 term of H.P. can be zero.
- If H is the H.M. between a and b, then

### Harmonic Progression Problems with Solutions

**1.**

**Solution:**

**2.**

**Solution:**

**3.**

**Solution:**

**4.**

**Solution:**

**5.**

**Solution:**

**6.**

**Solution:**

**7.**

**Solution:**