ILL 1984 Problem 31

Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$ . From the equation $f_1(x) = 0$ , one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$ . Continuing this process, we get equations $f_3,\cdots, f_n$ . Prove that $$\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha$$