Question

Insert four harmonic means between $\frac{2}{3}$ and $\frac{2}{13}$.

Answer 1

Let $a=\frac{2}{3}$ and $b=\frac{2}{13}$

Let $H_1,H_2,H_3,$ and $H_4$ are the required $4$ $HM$

$\therefore \frac{3}{2},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{H_3},\frac{1}{H_4},\frac{13}{2},$ are in $A.P.$

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

$\therefore \frac{13}{2}=T_{n+2}=\frac{3}{2}+(n+1)d$

$⟹d=\frac{\frac{13}{2}-\frac{3}{2}}{(n+1)}=\left(\frac{5}{n+1}\right)$

For $n=4$

$d=\frac{5}{4+1}=1⟹d=1$

$T_2=\frac{3}{2}+1=\frac{5}{2}=\frac{1}{H_1}$

$T_3=\frac{3}{2}+2\times 1=\frac{7}{2}=\frac{1}{H_2}$

$T_4=\frac{3}{2}+3\times 1=\frac{9}{2}=\frac{1}{H_3}$

$T_5=\frac{3}{2}+4\times 1=\frac{11}{2}=\frac{1}{H_4}$

$\therefore H_1=\frac{2}{5};$$H_2=\frac{2}{7};$$H_3=\frac{2}{9};$$H_4=\frac{2}{11}$