Proving 3^n >= 2n+1 by Induction

[It_Angel]

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
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]

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]

No, you can't just assert this -- you have to show it.

Use the fact that 3^(k + 1) = 3 * 3^k, and use your assumption that 3^i >= 2k + 1.

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.

 

[It_Angel]

Sorry, still at a loss.

 

[Mark44]

I already provided a strategy for you.

Use the fact that 3^(k + 1) = 3 * 3^k, and use your assumption that 3^i >= 2k + 1.

Did you not understand?