Harmonic Mean

If $p$ be the first of $n$ arithmetic means between two numbers and $q$ be the first of $n$ harmonic means between the same two numbers, prove that the value of $q$ cannot lie between $p$ and ${\left(\frac{n+1}{n-1}\right)}^{2}p.$

If $b$ is the harmonic mean between $a$ and $c,$ prove that $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$.

If $a, b, c$ are in $\mathrm{H}.\mathrm{P}.$, show that $a:\left(a-b\right)=\left(a+c\right):\left(a-c\right)$

If the ratio of harmonic mean to their geometric mean between two quantities is $12:13$. Prove that the quantities are in the ratio of $4:9$.

If $12$ and $9\frac{3}{5}$ are the geometric and harmonic means between two numbers respectively, find them.

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

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