Proof Using General principle of math induction

[kolley]

Homework Statement

prove n!>2^n for all n>=4

Homework Equations

The Attempt at a Solution

I showed it was true for n=1.

assume k!>2^k for all k>=4
then show it for k+1. (k+1)!>=2^(k+1)
=k!*(k+1)>=2*2^k
I don't know where to go from here.

 

[VeeEight]

I showed it was true for n=1.

But your base case is n=4!

Simplify 2*2^k

 

[kolley]

sorry, i meant for n=4

 

[Mark44]

assume k!>2^k for all k>=4
then show it for k+1. (k+1)!>=2^(k+1)
=k!*(k+1)>=2*2^k

This is what you need to show: (k+1)!>=2^(k+1).
In your next line, though, you are tacitly assuming that this is true. Also, don't connect one inequality to another with = as you are doing here:
(k+1)!>=2^(k+1)
=k!*(k+1)>=2*2^k

You have (k + 1)! = k! * (k + 1) = ? Here's where you use your induction hypothesis (i.e., k! >= 2^k).

 

[kolley]

I guess the factorial is what is throwing me off, I don't know how to use a chain of inequalities that will lead me to something that I can directly compare to 2^k+1 because I don't know how to take the factorial into account or get rid of it.

 

[VeeEight]

Use (k+1)! = (k+1) k! and apply the induction hypothesis