<s>[INST] Prove by induction that $2^1+2^2+2^3+2^4+ \cdots +2^n=2(2^n-1)$ Alright I have this problem,
Prove by induction $2^1+2^2+2^3+2^4+ \cdots +2^n=2(2^n-1)$
Now I've done this so far:
Base case $n=1$:
$$2^1 = 2$$
$$2(2^1-1)=2(2-1)=2(1)=2 .$$
Assume for $k$, prove for $k+1$:
$$
\begin{align*}
2^1+2^2+2^3+2^4+ \cdots +2^k+2^{k+1}

& =2(2^k-1)+2^{k+1}

\\ &=2^{k+1}+2^{k+1}-2
\end{align*}
$$
Now the trouble I'm running into is that I don't know how to continue from here, I know that I need to show that somehow $2^{k+1}+2^{k+1}-2 = 2(2^{k+1}-1)$.
Is there something that I would be missing with the rules of exponents, or maybe I just made a mistake and I'm doing something wrong that I don't recognize?
Any help would be appreciated, thanks.
 [/INST] $2^{k+1}+2^{k+1}-2 = 2\cdot 2^{k+1}-2 = 2(2^{k+1}-1)$
In first equality I use $a + a = 2a$, in second equality I factor out 2 (distributivity).
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