...For my proofwriting class, I have to show that $\displaystyle n! > a^{n}$ when $\displaystyle n \geq A $, with $\displaystyle a,n \in \mathbb{N}$. I know the eventual proof will be by induction, but I am not sure where to determine what the value of $\displaystyle A$. Messing around with the numbers 1-9, I was able to generalize it to two cases: If $\displaystyle a$ is even, then $\displaystyle A = \frac{5a-2}{2}$ and if $\displaystyle a$ is odd, then $\displaystyle A = \frac{5a-1}{2}$ , but is there a way to generalize what $\displaystyle A$ must equal for ANY natural number $\displaystyle a$?

Edit: Looking at $\displaystyle a > 9$, my generalization doesn't seem to hold. Now I am definitely confused.

Edit2: And just to clarify, I want $\displaystyle A$ to be the least possible value.