Re: Is this a legitimate proof by induction?

From: Steven (sgottlieb60_at_hotmail.com)
Date: 10/31/04 

Date: Sun, 31 Oct 2004 01:35:54 GMT

"Neil L." <imstr8trippin@yahoo.com> wrote in message
news:3qSdnXrPrYa9qxncRVn-vQ@rogers.com...
> Prove for all integers n: 2n + 4 <= (n + 2)!
>
> 1. For n = 1, 2(1) + 4 = 6 and (1 + 2)! = 6. Therefore, this formula is
> valid for n = 1.
> 2. For n = 2, 2(2) + 4 = 8 and (2 + 2)! = 24. Therefore, this is valid
for
> n = 2.
> 3. Assume n = k, 2k + 4 <= (k + 2)!. Prove that, 2(k+1) + 4 <= (k + 3)!
>
> R.S. = (k + 3)! = (k + 3)(k + 2)!
> (k + 3)(k + 2)! >= (k + 3)(2k + 4)
> (k + 3)(2k + 4) = 2k^2 + 10k + 12
> 2k^2 + 10k + 12 >= 10k + 12
> 10k + 12 >= 2k + 6
> 2k + 6 = 2(k + 1) + 4
>
> ?????????? This induction stuff is new to me, so any help/advice would be
> appreciated! Please point out anything I've done wrong/right. Thanks!
>
>
What you did is fine!
I would not have bothered with step 2
For step 3, I would just say---(k + 3)! = (k + 3)(k + 2)! >= (k + 3)(2k + 4)
= 2k^2 + 10k + 12 >= 10k + 12 >= 2k + 6= 2(k + 1) + 4
Steven

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