What are the solutions of the following exponential inequality for $n \in \mathbb{N}$?

Question

What are the solutions of the following exponential inequality for $n \in \mathbb{N}$?

$$n^3 < 4^{{(2n - 1)}^8}$$

I tried using WolframAlpha, but only an Inequality plot is returned, which I do not know how to interpret.

Answer 1

Prove it by induction. The general step is not hard. For $n \ge 2$:

$$(n+1)^3 < 8n^3 \le 8 4^{(2n-1)^8} = 4^{(2n-1)^8+2} \le 4^{(2(n+1)n-1)^8}$$

Checking $n=0,1,2$ directly shows the result true for all $n$.