Use mathematical induction to prove that the statement is true for every positive integer n.n! > 3n, for n ? 4

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). 4! > 3?4, so the statement is true for n = 4.
b). Assume the statement is true for n = k ? 4. Then,
 k! > 3k, or 
 (k)(k-1)...(1) > 3k.
 Multiply both sides by (k + 1):
 (k + 1)(k)(k-1)...(1) = (k + 1)! > 3(k+1)k
 Since k > 1, then (k + 1)k > (k + 1), so we can write
 (k + 1)! > 3(k + 1)k > 3(k + 1), or
 (k + 1)! > 3(k + 1)
The statement is true for n = k + 1 if it is true for n = k, and it is true for n = 4. Therefore, the statement is true for all n ? 4.