Question

Insert two harmonic means between $5$ and $11$.

Answer 1

Insert two $HM$ between $5$ and $11$

$\therefore a=5$ and $b=11$

Let $H_1$ and $H_2$ be the two $HM$

$\therefore \frac{1}{5},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{11}$ are in $A.P.$

Let $d$ is the common difference of this in $A.P$

$\therefore \frac{1}{11}={(n+2)}^{th}$ term $=T_{n+2}$

$⟹\frac{1}{11}=\frac{1}{5}+((n+2)-1)d$

$d=\frac{\frac{1}{11}-\frac{1}{5}}{(n+1)}$

$\therefore$ For $n=2$

$d=\frac{(\frac{1}{11}-\frac{1}{5})}{3}=\frac{5-11}{55\times 3}=\frac{-6}{3\times 55}=\frac{-2}{55}$

$\therefore T_2=\frac{1}{5}+(2-1)\times \frac{-2}{55}=\frac{11-2}{55}=\frac{9}{55}=\frac{1}{H_1}$

$T_3=\frac{1}{5}+(2\times \frac{-2}{55})=\frac{7}{55}=\frac{1}{H_2}$

$\therefore H_1=\frac{55}{9}$$H_2=\frac{55}{7}$