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MitsuZero
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Homework Statement
Original Problem Text:
http://img264.imageshack.us/img264/5882/problem3pn7.th.gif
Prove that, if n [tex]\in[/tex] [tex]N[/tex] and n [tex]\geq[/tex] 2, then 2^(n^2) > n^(n).
Homework 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.
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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