<s>[INST] How to prove a formula for the sum of powers of $2$ by induction? How do I prove this by induction?

Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$

Here is my attempt.
Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true.
Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$ 
Our hypothesis is: $2^n = 2^{n+1} -1$
Here is where I'm getting off track. Lets look at the right side of the last equation: $2^{n+1} -1$ I can rewrite this as the following.
$2^1(2^n) - 1$ But, from our hypothesis $2^n = 2^{n+1} - 1$ Thus:
$2^1(2^{n+1} -1) -1$ This is where I get lost. Because when I distribute through I get this.
$2^{n+2} -2 -1$ This is wrong is it not? Am I not applying the rules of exponents correctly here? I have the solution so I know what I'm doing is wrong. Here is the correct proof.

 [/INST] let $S = 2^0 + 2^1 + 2^2 + .... + 2^{n-1}$
so $2S = 2^1 + 2^2 + 2^3 + 2^n$
then $2S - S = S = (2^1 - 2^0) + (2^2 - 2^1) + ... + (2^{n-1} - 2^{n-1}) + 2^n = 2^n - 1$
and we got $2^0 + 2^1 + 2^2 + .... + 2^{n-1} = 2^n - 1$
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