Harmonic mean, Mathematics

If a, b and c are in harmonic progression with b as their harmonic mean then,

= 2003_harmonic mean.png

This is obtained as follows. Since a, b and c are in harmonic progression, 1/a, 1/b and 1/c are in arithmetic progression. Then,

        2272_harmonic mean1.png

This can be written as

1142_harmonic mean2.png

On cross multiplication we obtain

         2ac=b(a + c)

That is, b = 263_harmonic mean3.png

The second proposition we are going to look at in this part is: If A, G and H are the arithmetic, geometric and harmonic means respectively between two given quantities a and b then G2 = AH. The explanation is given below.

We know that the arithmetic mean of a and b is  605_harmonic mean4.png and it is given that this equals to A.

Similarly G2 = ab and H  = 1894_harmonic mean5.png  
The product of AH = 1369_harmonic mean6.png = ab. This we observe is equal to G2.

That is, G2 = AH, which says that G is the geometric mean between A and H.

Example 1.5.12

Insert two harmonic means between 4 and 12.

We convert these numbers into A.P. They will be 1/4 and 1/12. Including the two arithmetic means we have four terms in all. We are given the first and the fourth terms. Thus,

         T0      =       a = 1/4 and

         T4      =       a + 3d = 1/12

Substituting the value of a = 1/4 in T4, we have 

         1/4 + 3d     = 1/12

         3d             = 1/12 - 1/4 = - 1/6

         d               = -1/18

Using the values of a and d, we obtain T2 and T3.

         T2      =       a + d = 1/4 + (-1/18)

                                     = 1/4 - 1/18 = 7/36

         T3      =       a + 2d =  1/4 + 2.(-1/18)

                                      =  1/4 - 2/18

                                      =  1/4 - 1/9

                                      =   5/36

The reciprocals of these two terms are 36/7 and 36/5.

Therefore, the harmonic series after the insertion of two means will be 4, 36/7, 36/5 and 12.

Posted Date: 9/13/2012 4:49:27 AM | Location : United States

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