I have the suggestion that the following estimate is true for all $k\geq 2$:
$\frac{4}{n}\sum\limits_{i=1}^{k-1}\left(\frac{4}{n}+1\right)^i\leq \left(\frac{4}{n}+1\right)^{k+1}+(k+1)$
I tryed a lot, but couldn't solve it yet. Do you have an idea? If it helps, i can post my approaches.
Thank you!