Stuck on Proof by induction of 2^n>n^3 for all n>=10

[ArthurRead]

Homework Statement

Using the principle of mathematical induction, prove that for all n>=10, 2^n>n^3

Homework Equations

2^(n+1) = 2(2^n)
(n+1)^3 = n^3 + 3n^2 + 3n +1

The Attempt at a Solution

i) (Base case) Statement is true for n=10
ii)(inductive step) Suppose 2^n > n^3 for some integer >= 10
(show that 2^(n+1) > (n+1)^3 )
Consider 2^(n+1).
2^(n+1)= 2(2^n) > 2(n^3) = n^3 + n^3
(Ok, so this is where I'm stuck. Can I say n^3 > 3n^2 + 3n +1 because n>=10? Because if I can say that, then I can proceed with n^3 + n^3 > n^3 + 3n^2 + 3n +1 = (n+1)^3. I just don't know if i have to further justify it. Should I do another proof by induction to show that n^3 > 3n^2 + 3n +1 for n>=10? Or can I make a general statement that the power of 3 is higher than a power of 2 and so on)

Thank you!

 

[fresh_42]

3n^2 + 3n + 1 < 3n^2 + 3n^2 + 3n^2 = 9n^2 < 10n^2 <= n^3 should do

 

[HallsofIvy]

n^3- 3n^2+ 3n- 1 has derivative 3n^2- 6n+ 3= 3(n^2- 2n+ 1)= 3(n- 1)^2. That has a single 0 at n= 1< 10. Further at n= 10, 3(n-1)^2= 3*81> 0 so n^3- 3n^2+ 3n- 1 is increasing for all n> 10 and when n= 10, its value is 1000- 300+ 30+ 1= 729> 0 so n^3- 3n^2+ 3n- 1> 0 for all n> 10. Yes, n^3> 3n^2- 3n+ 1 for n> 10.

 

[ArthurRead]

Wow. That's creative. So basically I can't just say that n^3 > 3n^2 + 3n + 1 because 3n^2 isn't the only term? And is there a reason your last inequality is 10n^2 <= n^3 instead of just 10n^2 < n^3? Can I just say 10n^2 <n^3 because n>=10?

Thank you!

 

[fresh_42]

Wow. That's creative. So basically I can't just say that n^3 > 3n^2 + 3n + 1 because 3n^2 isn't the only term? And is there a reason your last inequality is 10n^2 <= n^3 instead of just 10n^2 < n^3? Can I just say 10n^2 <n^3 because n>=10?

Thank you!

I first had 10n^2 < n^3 but this is not true for the possible n=10. So either you state n>10 (which can be done in the induction step) or you just write the line from right to left. One "<" is enough.