Who is asking: Teacher
Level: All

Question:
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.

Penny

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