Question from John, a student:
I've been asked to prove this:
2^n > n^2 for n> 4 and n is a natural number
But I can't get past the part. So
(i) let n = 5, then 2^5 > 5^2
32 > 25 , true.
(ii) assume statement is true for n, prove it also holds for n+1
then I have 2^(n+1) > ( n+1)^2
I put it in the form 2^n * 2 > n^2 + 2n + 1
but I don't know how to use induction to make use of my assumption to prove this statement.
Any help would be appreciated.
Suppose we know k2 < 2k for some k ≥ 5, then (k+1)2 = k2(1+1/k)2 < 2k(1+1/5)2 =2k(36/25) < 2k+1.