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