Question

Three numbers are in harmonic progression. The harmonic mean between first and third numbers is $20$. If the $1^{st}$ number is twice the third number, find the three terms of the progression.

Answer 1

Let the three terms in harmonic progression are $x,y,z$
$H=\frac{2ab}{a+b}$
Here $a=x=2z$
$\therefore b=z$
$\therefore H=y=20$
$\therefore H=\frac{2\times 2z\times z}{2z+z}=\frac{4}{3}z$
$\therefore 20=\frac{4}{3}z$
$\therefore$ $4z=20\times 3$ $=60$
$\therefore z=\frac{60}{4}=15$
$\therefore$ $x=2z=2\times 15=30$
Therefore, the three terms are $30,20$ and $15$.