How to prove that $2^{n+1} = \Theta(2^n)$?

Question

I have a problem were I need to prove big theta. $f(n) = 2^{n+1} = Θ(2^n)$. I proved that this was true for big O but I'm not sure how to go about proving big Theta.

Answer 1

You need to show that there are positive constants $c_1,c_2$ such that

$$c_1 2^n \le 2^{n + 1} \le c_2 2^n$$ for all [sufficiently large] $n$.

For the $O$ you already found the second. The first is rather easier, so I am confident you will make it with this info at hand.

Answer 2

An other way to see it, is to find the limit

$$\lim_{n \to +\infty} \frac{2^{n+1}}{2^n}=\lim_{n \to +\infty} \frac{2^n \cdot 2}{2^n}=\lim_{n \to +\infty} 2=2$$

It is known that if $\lim_{n \to +\infty} \frac{f(n)}{g(n)}=c \in \mathbb{R}$ then $f(n)=\Theta(g(n))$.