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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)

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