Question 106
For each positive integer $n \ge 2$, define the polynomial $f_n(x)$ by $f_n(x) = x^n - x^{n-1} - x^{n-2} - \cdots - x - 1$. Prove that
(a) for each positive integer $n \ge 2$, the equation $f_n(x) = 0$ has a unique real positive root, say, $\alpha_n$;
(b) $(\alpha_n)_{n \ge 2}$ is a strictly increasing sequence; and
(c) $\lim_{n \to \infty} \alpha_n = 2$.
(a) for each positive integer $n \ge 2$, the equation $f_n(x) = 0$ has a unique real positive root, say, $\alpha_n$;
(b) $(\alpha_n)_{n \ge 2}$ is a strictly increasing sequence; and
(c) $\lim_{n \to \infty} \alpha_n = 2$.