Is 2^(2n) = O(2^n)

Question

Is 2(n+1) = O(2n)?

I believe that this one is correct because n+1 ~= n.

---

Is 2(2n) = O(2n)?

This one seems like it would use the same logic, but I'm not sure.

Answer 1

First case is obviously true - you just multiply the constant C in by 2.

Current answers to the second part of the question, look like a handwaving to me, so I will try to give a proper math explanation. Let's assume that the second part is true, then from the definition of big-O, you have:

which is clearly wrong, because there is no such constant that satisfy such inequality.