Geometric Series of $a = 2$, $r = 2$

Question

I'm having some trouble determining the geometric series for the following:

$$2^k + 2^{k-1} + \cdots + 2^2 + 2$$

With this, I'm guessing that $a = 2$, and common ratio is: $r = 2$

Most resources I find online just test for convergence. I'm looking for a summarized k formula.

Something $$2^k - 1$$

Answer 1

$2 + 2^2 + 2^3 + ... + 2^k = 2(1 + 2 + 2^2 + ... + 2^{k-1}) = 2\cdot \left(\dfrac{2^k - 1}{2-1}\right) = 2^{k+1} - 2$

Answer 2

If $S = 2+2^2+2^3+\dots +2^{k-1}+2^k$,

then $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2S=\,\,\,\,\,\,\,\,\,\,2^2+2^3+\dots +2^{k-1}+2^k+2^{k+1}$$$$\,\,\,S= 2+2^2+2^3+\dots +2^{k-1}+2^k$$

Hence $$S=2S-S=2^{k+1}-2$$