Question

Insert $3$ harmonic means between $2$ and $3$.

Answer 1

Let the harmonic series be

$\begin{array}{cc}\frac{1}{a},\frac{1}{a+2d},\frac{1}{a+2d},\frac{1}{a+3d}& \\ \frac{1}{a}=2& \\ \Rightarrow a=\frac{1}{2}& \left\{given\to \left(i\right)\right\}\\ and\frac{1}{a+3d}=3& \\ 1=3\left(a+3d\right)& \\ 1=3\left(\frac{1}{2}+3d\right)& \left\{from\left(i\right)\right\}\\ \frac{1}{3}-\frac{1}{2}=3d& \\ d=\frac{-1}{6\times 3}& \\ N{o}^{\prime }sare\frac{1}{\frac{1}{2}-\frac{1}{18}}and\frac{1}{\frac{1}{2}-\frac{1}{18}\times 2}& \\ \frac{18}{9-1}and\frac{18}{9-2}& \\ \frac{18}{8}and\frac{18}{7}& \\ \frac{9}{4}and\frac{18}{7}& \end{array}$