Ok I've just started a university maths course and the first thing we are learning is proofs. (I hope this is the right sub-forum )
I get the concept and all but I am reading a book on the subject anyway and it has a proof early on that I don't quite get the reasoning of.
Its to prove 2^n > n^2 for n>or=5.
I get that we have to prove that 2^(n+1) > (n+1)^2
So we multiply 2^n > n^2 X2
2X 2^n = 2^(n+1) > 2X n^2 = 2n^2 = n^2 + n^2 = n^2 +nn
Now this is the part that I don't quite understand. It has;
Since n> or =5, we have n> or =3 (Why 3? In using 3 there is an inequality in the wrong direction?) so
nn> or = 3n = 2n+n > or = 2n+1 (I don't see how you can just assume that the n can turn into a 1 either)
I see that this gets the answer to the needed 2^(n+1) > n^2 + 2n + 1 = (n+1)^2
but can't justify why you can just use the things pointed out above to get the reasoning.
Personally I would have argued (this is by no means a correct statement so if I'm gravely wrong please say) That n^2 + n^2 is always going to be greater than n^2 + 2n + 1 for n> or =5 as n^2 is always going to be larger then 2n + 1 for numbers that high. I know thats not quite induction but to me it makes more sense the swapping out n for convenient numbers.
Thanks for the help!