Question

Mathematical induction problem. Details in my first post to the problem.

Answer 1

Prove that for n >1, \[n ^{n} \le (n!)^{2}\] It seems like induction is the way to go on this, so I am assuming that, \[k ^{k} \le (k!)^{2}\] is true and then it needs to be shown that \[(k+1) ^{k+1} \le [(k+1)!]^{2}\] So, if we take \[k ^{k} \le (k!)^{2}\] and multiply both sides by (k+1)^2, we get \[(k+1)^{2}(k ^{k}) \le [(k+1)!]^{2}\] If I can somehow show that the left-hand side of the fifth equation, [(k+1)^2](k^k) is greater than the left-hand side of the third equation, (k+1)^(k+1), I will have completed the proof, so it seems to me that I basically have to show that\[(k+1)(k ^{k}) \ge (k+1)^{k}\] I don't seem to be having much luck using logs, and I don't think that expanding the right-hand side of this last equation is the way to go. So, this is as far as I got, which should show some effort. Can anyone help with this last equation (assuming I'm right so far), and if I'm not, can anyone take it from the top?

Answer 2

@jim_thompson5910 @Algebraic!

Answer 3

@satellite73 @amistre @myininya

Answer 4

The more I think about it, I really believe that I have to use Pascal's Rule. I am almost positive about this.

Answer 5

Pascal's Rule.