Ok I can't figure out how to inductively prove the following inequalities, I have the base case and inductive hypothesis, I just don't see how to prove this stuff. I've already turned in the assignment, I'm just curious cause they never tell us. At least there's points in it for you guys.
1.) Prove that n! > n^3 when n is large enough. (n has to be greater than or equal to 6)
2.) Prove that n! < n^n for all positive integers by induction.
I just can't seem to prove these inequalities for P(K+1). Any help would be appreciated.
-Jeff P.S. (If you solve one I'll give you half the points, unless only one is ever solved than I'll give you them all! HAHAHA)
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First prove P(6), which is 6! > 6^3. This is just easy arithmetic.
Then you have to prove P(k+1) given P(k) and k >= 6.
Given k! > k^3, and k >= 6, prove that (k+1)! > (k+1)^3.
Have you gotten this far? The first thing you should do is expand the factorial and the polynomial.
2. n! < n^n for all n >= 1.
This is actually not true. 1! < 1^1 reduces to 1 < 1, which is false.
Maybe you meant n! <= n^n for all n >= 0, or maybe n! < n^n for all n >= 2.
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Assuming that k! < k^k, what can you then say about (k+1)! and (k+1)^(k+1) ?
what do you know about 1! and 1^1