Using fields or triangle inequalities, show that 2n Solution n!>2^n and n > 4 In

Question

Using fields or triangle inequalities, show that

Explanation / Answer

n!>2^n and n > 4 In class the proof might look something like this: (n+1)!=n! (n+1) from the inductive hypothesis we have n! (n+1) > 2^n (n+1) since n>1 we have 2^n(n+1) > 2^n(2) and 2^n(2) = 2^{n+1} Now, we can string it all togther to get the inequality: (n+1)!=n! (n+1) > 2^n(n+1) > 2^n(2) =2^{n+1} (n+1)! > 2^{n+1} In general, it's worth trying to figure out wether it 'safe' to multiply n!>2^n by n+1 > 2 while preserving the inequality. Hope this helps :)

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