MHB Proof of Inequality by Induction

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The discussion focuses on proving the inequality n^3 ≤ 3^n for all natural numbers n using mathematical induction. Participants are seeking assistance with the proof, emphasizing the need for a clear induction step. A hint is provided, suggesting that for n ≥ 3, the expression (n+1)^3 can be compared to (n + n/3)^3 to facilitate the proof. The conversation highlights the importance of establishing the base case and the inductive step to complete the proof successfully. The overall goal is to demonstrate the validity of the inequality through rigorous mathematical reasoning.
sbrajagopal2690
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need help on this

Show by induction that n^3 <= 3^n for all natural numbers n.
 
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Re: Proof of Inequlity by Induction

sbrajagopal2690 said:
need help on this

Show by induction that n^3 <= 3^n for all natural numbers n.
Hint: If $n\geqslant 3$ then $(n+1)^3 \leqslant \bigl(n+\frac n3\bigr)^3$.
 

Thread 'Two interesting number theory tricks'

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