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

For a system of homogeneous linear equations to have a non-zero solution, the determinant of the coefficients must be zero:
$$\left|\begin{array}{ccc}1& 2a& a\\ 1& 3b& b\\ 1& 4c& c\end{array}\right|=0$$
Expanding along the first row:
$$1(3bc-4bc)-2a(c-b)+a(4c-3b)=0$$
$$-bc-2ac+2ab+4ac-3ab=0$$
$$2ac-ab-bc=0$$
$$2ac=b(a+c)$$
$$b=\frac{2ac}{a+c}$$
This is the condition for $a,b,c$ to be in Harmonic Progression (H.P.).