# JEE Main & Advanced Mathematics Other Series Harmonic Mean

## Harmonic Mean

Category : JEE Main & Advanced

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.

Also, if a, H, b are in H.P., then H is called harmonic mean between $a$ and $b$.

(1) Insertion of harmonic means

(i) Single H.M. between $a$ and $b$$=\frac{2ab}{a+b}$.

(ii) H, H.M. of $n$ non-zero numbers ${{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,....,\,{{a}_{n}}$  is given by $\frac{1}{H}=\frac{\frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+.....+\frac{1}{{{a}_{n}}}}{n}$.

(iii) Let $a,\,\,b$ be two given numbers. If $n$ numbers ${{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}$ are inserted between $a$ and $b$ such that the sequence  $a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b$ is a H.P., then ${{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}$ are called $n$ harmonic means between $a$ and $b$.

Now, $a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b$ are in H.P.

$\Rightarrow$ $\frac{1}{a},\,\frac{1}{{{H}_{1}}},\,\frac{1}{{{H}_{2}}},\,......\frac{1}{{{H}_{n}}},\,\frac{1}{b}$ are in A.P.

Let $D$ be the common difference of this A.P. Then,

$\frac{1}{b}={{(n+2)}^{th}}\text{ term }={{T}_{n+2}}$

$\frac{1}{b}=\frac{1}{a}+(n+1)\,D$$\Rightarrow$$D=\frac{a-b}{(n+1)\,ab}$.

Thus, if $n$ harmonic means are inserted between two given numbers $a$ and $b,$ then the common difference of the corresponding A.P. is given by $D=\frac{a-b}{(n+1)\,ab}$.

Also, $\frac{1}{{{H}_{1}}}=\frac{1}{a}+D$, $\frac{1}{{{H}_{2}}}=\frac{1}{a}+2D$,…….,$\frac{1}{{{H}_{n}}}=\frac{1}{a}+nD$,

where $D=\frac{a-b}{(n+1)\,ab}$.

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