The discussion focuses on proving the inequality \( n^3 \leq 3^n \) for all natural numbers \( n \) using mathematical induction. Participants emphasize the importance of establishing a base case and an inductive step, particularly for \( n \geq 3 \). A hint is provided suggesting that if \( n \geq 3 \), then \( (n+1)^3 \leq \left(n + \frac{n}{3}\right)^3 \) can be utilized to facilitate the proof. This structured approach confirms the validity of the inequality through rigorous mathematical reasoning.
PREREQUISITESStudents in mathematics, educators teaching proof techniques, and anyone interested in deepening their understanding of mathematical induction and inequalities.
Hint: If $n\geqslant 3$ then $(n+1)^3 \leqslant \bigl(n+\frac n3\bigr)^3$.sbrajagopal2690 said:need help on this
Show by induction that n^3 <= 3^n for all natural numbers n.