Let $a,b \in \mathbb{R}$ and $n \in \mathbb{N} $|$\frac {a^{n+1} - b^{n+1}}{a^{n} - b^{n}} - a |\leq \frac{1}{100}$ How can i finde a n for which this inequality is true ?
PS:
$a = \frac{1+ \sqrt{5}}{2}, b =\frac{1 - \sqrt{5}}{2} $. And $f_n$ is the Fibonacci sequence and $b_n := \frac{f_{n+1}}{f_n}$. So $b_n$ converges to the golden ratio and i want to finde a n such that $b_n$ is closer to the golden ratio than $\frac{1}{100}$