Lesson 1

Using the method of mathematical induction, one can prove statements that are true not for all positive integers, but for all integers \(n \ge \alpha\) (for a fixed integer \(\alpha\)). To do this, one ensures that the base case \(n=\alpha\) holds and that the step \(n \to n+1\) holds for any integer \(n \ge \alpha\).

Problem. It is possible to show that \(2^{n} \ge n^{3}\) for all sufficiently large integers \(n\). Find the minimum integer \(\alpha\) such that \(2^{n} \ge n^{3}\), for all integers \(n \ge \alpha\).