Find $n$ such that $1-a c^{n-1} \ge \exp(-\frac{1}{n})$ https://ift.tt/eA8V8J

I am trying to find the integer $n$ such that \begin{align} 1-a c^{n-1} \ge \exp(-\frac{1}{n}) \end{align} where $a>0$ and $c \in (0,1)$.
I know that finding it exactly is difficult. However, can one find good upper and lower bounds it.

It tried using lower bound $\exp(-x) \le 1-x+\frac{1}{2}x^2$. However, it didn't really work.

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Lisa