## Algebra

Sequence and Series - 3

**Harmonic Progression:**

If numbers are in Harmonic Progression or H.P., then their reciprocals are in A.P. So, if there are three numbers a, b and c in Harmonic Progression, then their reciprocals i.e. 1/a, 1/b and 1/c are in Arithmetic Progression.

For example, consider the numbers 30,40 and 60

2/40 = 1/30 + 1/60

So, these three numbers are in H.P.

Harmonic Mean of two numbers:

For two numbers a and b, their Harmonic Mean or H.M. is given by 2ab/(a+b) or

**Harmonic Mean of n numbers:**

**Example 1**: If the harmonic mean between two quantities is to their geometric means is 12 to 13. Prove that the quantities are in the ratio of 4 to 9.

**Solution:** Let the two numbers be a and b respectively.

Their geometric mean = √ab and their harmonic mean = 2ab/(a+b)

12/13 = 2√ab/(a+b)

or (a+b)/2√ab = 13/12

Using componendo and dividend, we get:

**or a/b = 4/9**

**Example 2:** If b is the harmonic mean between a and c, prove that

1/(b-a) + 1/(b-c) = 1/a + 1/c

**Solution:** Given that b is the harmonic mean between a and c, so

b = 2ac/(a+c)