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

The system will have a non-zero solution ifΔ=12aa13bb14cc=0Using C2→C2-2C3, we get Δ=10a1bb12cc=0⇒1(bc-2bc)+a(2c-b)=0⇒2ac=bc+ab⇒2aca+c=b⇒a,b,c are in H.P.