> Is $2^{n+1} = O(2^{n})$ ?
- Yes.
- Reason : $\space 2 \cdot 2^{n} \le c \cdot 2^{n}$ where $c$ is a positive real number, $c \ge 2$ in this particular case.
> Is $2^{2n} = O(2^{n})$ ?
- No.
- Reason :
$2^{n+n} \le c \cdot 2^{n} $ where c is a positive real number. There's no such constant $c$ that satisfies this inequality.