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

Related Discussions:- Harmonic mean, Assignment Help, Ask Question on Harmonic mean, Get Answer, Expert's Help, Harmonic mean Discussions

Write discussion on Harmonic mean
Your posts are moderated
Related Questions
Give the Introduction to amino ACID and nucleotide metabolism ? Here, we studied about the chemistry of proteins and amino acids. We studied that the amino acids are used for p

At the school bookstore and two binders and three pens cost $12.50. Three binders and five pens cost $19.50. What is the approximate cost of 1 binder and 1 pen? Let x = the cos

Why -2=-x , is x=2

Write a program to find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. The area under a curve between two points can b

Find out the volume of the solid obtained by rotating the region bounded by x =  (y - 2) 2 and  y = x around the line y = -1. Solution : We have to first get the intersection

Alternate Notation : Next we have to discuss some alternate notation for the derivative. The typical derivative notation is the "prime" notation. Though, there is another notation

want to make an assignment on shares and dividend for class 10

How to Converting Scientific Notation to Standard Notation ? To change a number in scientific notation to standard notation, move the decimal point the same number of places as