(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Original Problem Text:

http://img264.imageshack.us/img264/5882/problem3pn7.th.gif [Broken]

Prove that, if n [tex]\in[/tex] [tex]N[/tex] and n [tex]\geq[/tex] 2, then 2^(n^2) > n^(n).

2. Relevant equations

Some Hints:

*** (n-1)^(n-1) * 2^(n-1) = (2n - 2)^(n-1)

*** If n >= 2 then (2n - 2) >= n.

*** If n >= 1 then 2^(n) >= n.

3. The attempt at a solution

Using Induction:

For the base step, suppose that n = 2. This implies that 2^(2^2) = 16, and 2^(2)= 4, and 16 > 4, so the base step is proven true.

Suppose that n>2, and that the theorem has been proven for all smaller values of n.

Then we compute:

2^(n^2) > n^n

2^(n-1)^2 > (n-1)^(n-1)

***Multiply both sides by 2^(n-1)***

2^(n-1)^2 * 2^(n-1) > (n-1)^(n-1) * 2^(n-1)

2^(n-1)^2 * 2^(n-1) = 2^(n^2 - n) = 2^(n(n-1))

2^(n(n-1)) > (2n -2)^(n-1)

***Now here's the screwy part - I took the (n-1) root across the inequality. I have my doubts about the mathematical legality of such an operation. ***

2^(n) > 2n - 2

2n -2 [tex]\geq[/tex] n (since n [tex]\geq[/tex] 2)

2^(n) > n

2^(n^2) > n^(2)

Does this proof make any sense at all? Does it even remotely resemble a proof? I should point out that I'm using (n-1) in place of (n+1) because that's the way my professor did it.

Thanks a lot for any help!

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# Homework Help: Proof through Induction with Inequality

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