Quandaries and Queries


Name: Jose

Who is asking: Teacher
Level: All

How can you prove by mathematical induction that:

n! > n2.

I will appreciate your help.



Hi Jose,

Let P(n) be the statement that n! > n2.

P(1) is false.
P(2) is false.
P(3) is false.
P(4): 4! =24 > 16 = 42 is true.

Assume that you have checked out the validity of the statement P(n) for n = 4, 5, ..., k for some k > 4.

Consider P(k+1):

(k+1)! = (k+1)k! > (k+1)k2 by our induction hypothesis.

Now if we had (k+1)k2 = k3 +k2 greater than (k+1)2 for k > 4 we'd be finished. There are a number of ways of showing this - you can even do it inductively.

Another tact would be to write (k+1)! = (k+1)k! and then show, by induction, that k! > k+1.

Good luck with it.



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