They call it Progressions?

$\small{a,b,c \text{ are in H.P}}\implies b=\dfrac{2ac}{a+c}... (1)$

$\small{a^2,b^2,c^2 \text{ are in A.P}}\implies b^2=\dfrac{a^2+c^2}2...(2)$

Squaring $(1)$ and then comparing with $(2)$ ,

$\begin{aligned}&\dfrac{4a^2c^2}{(a+c)^2}=\dfrac{(a^2+c^2)}2\\\implies&\dfrac{4\frac{c^2}{a^2}}{(1+\frac ca)^2}=\dfrac{(1+\frac{c^2}{a^2})}2\\\implies&\dfrac{8t^2}{(1+t)^2}=1+t^2~~\small{\left(t=\dfrac ca\right)}\end{aligned}$

$\implies t^4+\cdots\cdots +1=0$

By Vieta's formula product of roots is $\large\color{#0C6AC7}{\boxed 1}$ .