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

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,
This can be written as
On cross multiplication we obtain
2ac=b(a + c)
That is, b 
= 

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 G^{2} = AH. The explanation is given below.
We know that the arithmetic mean of a and b is and it is given that this equals to A.
Similarly G^{2} = ab and H = 


The product of AH = 

= ab. This we observe is equal to G^{2}. 
That is, G^{2} = 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,
T_{0} = a = 1/4 and
T_{4} = a + 3d = 1/12
Substituting the value of a = 1/4 in T_{4}, 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 T_{2} and T_{3}.
T_{2} = a + d = 1/4 + (1/18)
= 1/4  1/18 = 7/36
T_{3} = 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.