Question

Insert 2 harmonic mean between 7 and 8.

• A. $22,23$
• B. $\frac{168}{22},\frac{168}{20}$
• C. $\frac{168}{23},\frac{168}{22}$
• D. $9,10$

Answer 1

Answer: (C)

$\frac{168}{23},\frac{168}{22}$

$\Rightarrow$ Let $a=7,b=8$ be two given numbers and $H_{1}and{H}_2$ are $2$ harmonic mean between $a$ and $b$.

$\Rightarrow$ Now, $a,H_1,H_2,b$ are in $H.P.$

$\Rightarrow$ $\frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$ are in $A.P$.

$\Rightarrow$ Let $D$ be the common difference of this $A.P.$

$\Rightarrow$ $D=\frac{a-b}{(n+1)ab}=\frac{7-8}{(2+1)\times 7\times 8}=\frac{-1}{168}$

$\Rightarrow$ Now, $\frac{1}{H_1}=\frac{1}{a}+D=\frac{1}{7}+\left(\frac{-1}{168}\right)=\frac{23}{168}$ ...... $\left(1\right)$

$\Rightarrow$ $\frac{1}{H_2}=\frac{1}{a}+2D=\frac{1}{7}+2\times \left(\frac{-1}{168}\right)=\frac{22}{168}$ ...... $\left(2\right)$

Hence from $\left(1\right)$ and $\left(2\right)$, we get $H_1=\frac{168}{23},H_2=\frac{168}{22}.$