Proof by induction of the formula for $2^0+2^1+2^2+...+2^n$?

Question

$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N} \cup \{0\}$.

I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction.

---

I tested the base case where it's equal to zero, and it worked.

Then I did "assume $k∈ \mathbb{N} \cup \{0\}$ and $2^0+2^1+2^2+...+2^k$". I got marked down for saying "Let $n=k+1$", and did a bunch of math for it to become $2^{k+2} -1$ and I was apparently wrong >_>?

Answer 1

You have to prove it on both sides, but you can make the assumption that you prove it for earlier cases.

So to prove for $n+1$ you may assume it holds for $n$.