I am new to proofs and I am trying to learn mathematical induction. I started working out a sample problem, but I am not sure if I am on the right track. I was wondering if someone would be kind enough to comment on my work so far, and give me some hints as to how I should proceed.Many thanks in advance!

$S_n$ is the minimum number of moves it takes to solve towers of Hanoi where $n$ is a positive integer.

$$S_n = 2^n-1$$

**Base Case:**
$$\begin{align*}S_1 &= 2^1-1 \\&= 1 \end{align*}$$

Assume true for $k$:

$$S_k = 2^k-1$$

Hence,

$$\begin{align*}S_{k+1} &= (2^1-1) , (2^2-1) , (2^3-1) , \ldots , (2^k-1) , (2^{k+1}-1) \\&= (2^k-1) + 2^{k+1}-1\end{align*}$$ This is where I am stuck.