If the system of linear equations $x+2ay+az=0$, $x+3by+bz=0$ and $x+4cy+cz=0$ has a non-zero solution, then:

For a homogeneous system of equations to have a non-zero solution, the determinant of the coefficient matrix must be zero ($\Delta =0$).
$$\left|\begin{array}{ccc}1& 2a& a\\ 1& 3b& b\\ 1& 4c& c\end{array}\right|=0$$
Applying column operation $C_2\to C_2-2C_3$:
$$\left|\begin{array}{ccc}1& 0& a\\ 1& b& b\\ 1& 2c& c\end{array}\right|=0$$
Expanding along the first row:$1(bc-2bc)-0+a(2c-b)=0$$-bc+2ac-ab=0$$2ac=ab+bc$$2ac=b(a+c)$$b=\frac{2ac}{a+c}$ This indicates that $a,b,c$ are in Harmonic Progression (HP).