Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Harmonic Progression (H.P.) & Harmonic Mean (H.M.)

A number of terms  a_1,a_2...,a_n are said to be in Harmonic Progression (H.P) if their reciprocals  \dfrac{1}{a_1},\dfrac{1}{a_2},...\dfrac{1}{a_n} are in A.P.

 a_n , the  n^{th} term of the H.P, is given by  a_n=\dfrac{1}{a=(n-1)}d , where  a=\dfrac{1}{a_1} and  d=\dfrac{1}{a_2}-\dfrac{1}{a_1}

Harmonic Mean (H.M)

If  a_1,a_2...,a_n are in H.P, then their Harmonic Mean (H.M) is given by

 \dfrac{1}{\text{H}}=\dfrac{1}{n}\left( \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_n} \right)

This means reciprocals of the Harmonic mean is the Arithmetic Mean of the reciprocals.

Some Facts about H.P -:

(i) If  a and  b are two non-zero numbers, then the harmonic mean of  a and  b is number.

 \text{H} such that the sequence  a, \text{H}, b is a H.P.

We have  \dfrac{1}{\text{H}}=\dfrac{1}{2} \left( \dfrac{1}{a} +\dfrac{1}{b}\right) or  \text{H}=\dfrac{2ab}{a+b}

(ii) If  a_1,a_2...,a_n are non-zero numbers, then the harmonic mean  \text{H} of these numbers is given by   \dfrac{1}{\text{H}}=\dfrac{1}{n}\left( \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_n} \right)

(iii) The  n numbers  \operatorname{H_1,H_2,…,H_n} are said to be harmonic means between  a and  b , if  \operatorname{a_1,H_1,H_2,…,H_n,b}  are in H.P, i.e. if  \dfrac{1}{a}, ,\dfrac{1}{\operatorname{H_1}},\dfrac{1}{\operatorname{H_2}},...,\dfrac{1}{\operatorname{H_n}},\dfrac{1}{b} are in A.P.

Let  d be the common difference of this AP.

Then  \dfrac{1}{b}=\dfrac{1}{a}+(n+2-1)d

 \implies d=\dfrac{1}{n+1}\left( \dfrac{1}{b}-\dfrac{1}{a} \right)=\dfrac{a-b}{(n+1)ab}

\text{Thus } \begin{array}{ccc} \dfrac{1}{\operatorname{H_1}}=\dfrac{1}{a}+\dfrac{a-b}{(n+1)ab}\\ \dfrac{1}{\operatorname{H_2}}=\dfrac{1}{a}+\dfrac{2(a-b)}{(n+1)ab}\\\vdots \quad \quad \vdots~~~~~~~~~~~~~~~~~~\\\dfrac{1}{\operatorname{H_n}}=\dfrac{1}{a}+\dfrac{n(a-b)}{(n+1)ab} \end{array}

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