difficulty following proof by induction

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November 17th 2011, 11:29 AM
Jskid

difficulty following proof by induction

This is given as an example of proof by induction. Prove that $n! > 2^n$ for all n >= 4

$n_0=4$ and $4!=24>16=2^4$ . Thus $n_0$ is true. Now suppose that k >= 4 and the statement is true for n=k. Thus we suppose $k! > 2^k$ We must prove that the statement is true for n=k+1; that is, we must prove that $(k+1)! > 2^{k+1}$ . Now
$(k+1)!=(k+1)k!>(k+1)2^k$
using the induction hpothesis. Since k >= 4 certainly k+1 > 2 so $(k+1)2^k>2 x 2^k = 2^{k+1}$ . We conclude that $(k+1)! > 2^{k+1}$ as desired. By the principle of mathematical induction we conclude that $n!>2^n$ for all integers n>=4

I don't get the part after "certainly k+1>2". What happened to the factorial?
November 17th 2011, 12:17 PM
DrSteve

Re: difficulty following proof by induction

It doesn't say that. It says k+1>2 which it is because k is at least 4. In fact, k+1>4, but all that's needed for the proof is k+1>2.

Also you have a typo. That minus sign towards the end should be an equal sign.
November 17th 2011, 08:13 PM
Jskid

Re: difficulty following proof by induction

Ok but how does that show anything about (k+1)!?
November 17th 2011, 08:25 PM
Prove It

Re: difficulty following proof by induction

Quote:

Originally Posted by Jskid

Ok but how does that show anything about (k+1)!?

You know $\displaystyle (k + 1)! = (k + 1)k! > (k + 1)2^k$ and you have shown $\displaystyle (k + 1)2^k > 2^{k + 1}$ .

Therefore $\displaystyle (k + 1)! > 2^{k + 1}$ .