MHB Proof of Inequality by Induction

Click For Summary
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
Messages
2
Reaction score
0
need help on this

Show by induction that n^3 <= 3^n for all natural numbers n.
 
Physics news on Phys.org
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 'A variant of the Monty Hall problem'

There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...
View full post »

Similar threads

MHB Proof by induction - Really confused
  • · Replies 4 ·
Replies
4
Views
2K
MHB Understanding induction proof of pigeonhole principle
  • · Replies 4 ·
Replies
4
Views
4K
A problem of mathematical induction
  • · Replies 8 ·
Replies
8
Views
2K
MHB Can We Prove the Inductive Set Contains Only Natural Numbers?
  • · Replies 10 ·
Replies
10
Views
2K
MHB Prove 1/√n Inequality by Induction
  • · Replies 1 ·
Replies
1
Views
2K
MHB Proof by Induction - in Sequences.
  • · Replies 2 ·
Replies
2
Views
2K
MHB Double Induction: Proving Addition is Commutative
  • · Replies 6 ·
Replies
6
Views
2K
MHB Sequence of b_{k} with Explicit Formula: Proving by Math Induction
  • · Replies 2 ·
Replies
2
Views
1K
MHB Proof by induction? No Idea what I should do :(
  • · Replies 4 ·
Replies
4
Views
1K
MHB Math induction with sigma notation
  • · Replies 5 ·
Replies
5
Views
2K