OME 1987 Problem 6

For all natural numbers $n$ , consider the polynomial $P_n(x) = x^{n+2}-2x+1$ . (a) Show that the equation $P_n(x)=0$ has exactly one root $c_n$ in the open interval $(0,1)$ . (b) Find $lim_{n \to \infty}c_n$ .