- #1
ArthurRead
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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!