Help with a recursively defined sequence
So my problem is as follows:
I have a recursively defined sequence with a(1)=5, and a(n+1)=a(n)^2-a(n). The problem is to show that a(n)>2^2^n for all n>=1.
Proof by Induction:
Base: a(1) = 5 > 4 = 2^2^1, so it's true for n=1.
Induction: a(n+1)=a(n)^2-a(n)>(2^2^n)^2-(2^2^n) and this is where I get stuck. Try as I might I can't seem to show that a(n+1)>2^2^(n+1).
Any help will be much appreciated, and I'm not looking for a full solution just some hints would be great. Thanks