Question 185
Let $a \gt 2$ be a real number and $f_n(x) = a^{10} x^{n+10} + x^n + \dots + x + 1$ ($n = 1, 2, \dots$). Prove that for every positive integer $n$ the equation $f_n(x) = a$ has exactly one real root $x_n \in (0; +\infty)$. Prove that the sequence $(x_n)$ has a finite limit when $n \to +\infty$.