The problem involves proving the inequality 3^n >= 2n + 1 for all natural numbers using mathematical induction.
Some participants are exploring the inductive step and questioning the assumptions made during the proof. Guidance has been offered regarding the use of the inductive hypothesis, but there is no explicit consensus on the correctness of the approach taken.
There is a focus on ensuring that each step of the induction is properly justified, with some participants expressing confusion about the reasoning used in the attempts. The discussion reflects a need for clarity in the proof structure.
No, you can't just assert this -- you have to show it.It_Angel said:Homework Statement
Prove that 3^n >= 2n+1 for all natural numbers.
Homework Equations
3^n >= 2n+1 [is bigger or equal to]
The Attempt at a Solution
3*1>=2+1
True for n=1
Assumption: 3^k>=2k+1
3^(k+1)>=2k+3[/color]
It_Angel said:3^k*3>=2k+3
(2k+1)*3>=2k+3 <---can I just substitute 2k+1 into 3^k as per my assumption, because 3^k is bigger, and (2k+1)*3 is bigger than 2k+3? If so, is the proof complete?
Thanks in advance.
Mark44 said:Use the fact that 3^(k + 1) = 3 * 3k, and use your assumption that 3^i >= 2k + 1.