Proving n2 < 2n using Mathematical Induction

[skeough15]

Homework Statement

We are asked to try and prove the values n where n² < 2⁽ⁿ⁾ .
It asks us to prove it by Math Induction.

The Attempt at a Solution

I can see it works for n=0 and n=1 but not for n=2,3,4 . So I made my base step n=5 and showed that 5² = 25 < 2⁵ =32 as 25<32. I then started simple mathematical induction off by assuming that n²<2ⁿ for n>5, and tried to prove it using (n+1)²<2⁽ⁿ⁺¹⁾ but can't seem to get that proven. Any suggested help on how to prove that?

 

[micromass]

It seems that you need to show that $$2n+1<2^n$$... Maybe another induction?

 

[skeough15]

yes that is exactly what I had to do, I assume it is sufficient to have an induction proof within an induction proof?

 

[micromass]

I can't really see a problem with that...

 

[silvermane]

For help, check out the thread:
https://www.physicsforums.com/showthread.php?t=232381