Question 7
Let $a\gt 2$ be a real number. Put
$$f_n(x)=a^{10}x^{n+10}+x^n+\cdots+x+1\qquad(n=1,2,\ldots).$$
Prove that for every $n$, the equation $f_n(x)=a$ has exactly one solution $x_n\in(0,+\infty)$. Prove that the sequence $(x_n)$ has a finite limit as $n\to+\infty$.
$$f_n(x)=a^{10}x^{n+10}+x^n+\cdots+x+1\qquad(n=1,2,\ldots).$$
Prove that for every $n$, the equation $f_n(x)=a$ has exactly one solution $x_n\in(0,+\infty)$. Prove that the sequence $(x_n)$ has a finite limit as $n\to+\infty$.