### Home > PC > Chapter 8 > Lesson 8.2.6 > Problem 8-152

Copy the steps below and fill in the blanks to prove that

*n*! > 2for^{n}*n*≥ 4. Note that we do not start with*n*= 1. Verify that the statement is false for*n*< 4. Homework Help ✎Verify the statement is true for

*n*= 4.Assume the statement is true for

*n*=*k*.Prove the statement is true for

*n*=*k*+ 1.Recall that: (

*k*+ 1)! = ____ (*k*+ 1)We assumed that

*k*! > 2, therefore (^{k}*k*+ 1)! = ____ (*k*+ 1) > ____ (*k*+ 1).Because

*k*≥ 4,*k*+ 1 > 2 and hence ____ (*k*+ 1) > 2^{k}^{ }· ____ = 2−.Following the steps from above, we have shown that ____ > ____.

Write a conclusion for your proof by induction.

4! = 24 > 2^{4} = 16 whereas 3! = 6 ≯ 2^{3} = 8.

Assume that *k*! > 2* ^{k}* for

*k*≥ 4.

Hence, by mathematical induction we have proven that *n*! > 2* ^{n}* for

*n*≥ 4.