Harmonic mean, Mathematics

Assignment Help:

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.


Related Discussions:- Harmonic mean

VAM, applications of VAM.

applications of VAM.

Trigonometry, If tanA+sinA=m and m2-n2 = 4vmn, show that tanA-sinA=n

If tanA+sinA=m and m2-n2 = 4vmn, show that tanA-sinA=n

Derivatives for logarithm, Logarithm Functions : Now let's briefly get the...

Logarithm Functions : Now let's briefly get the derivatives for logarithms.  In this case we will have to start with the following fact regarding functions that are inverses of ea

Find the tangent to the curve, 1. Find the third and fourth derivatives of ...

1. Find the third and fourth derivatives of the function Y=5x 7 +3x-6-17x -3 2. Find the Tangent to the curve Y= 5x 3 +2x-1 At the point where x = 2.

Use the power function to find derivative, Given, y = f(x) = 2 x 3 - 3x 2 ...

Given, y = f(x) = 2 x 3 - 3x 2 + 4x +5 a)  Use the Power function to find derivative of the function. b)  Find the value of the derivative at x = 4.

determine that the relation is symmetric and transitive, 1. Let R and S be...

1. Let R and S be relations on a set A. For each statement, conclude whether it is true or false. In each case, provide a proof or a counterexample, whichever applies. (a) If R

Parent, Sam has 18 marbles. Dean has 3 marbles. Dean has ---- as many marbl...

Sam has 18 marbles. Dean has 3 marbles. Dean has ---- as many marbles as Sam?

Assisnment, How a student of mathematics of b.sc can make an assignment on...

How a student of mathematics of b.sc can make an assignment on topic of asymptotes?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd