<s>[INST] Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$
I know the answer is $2^n - 1$, but how to simplify it?
 [/INST] $$\begin{eqnarray} 
\\ \text{Let } 2^{(n-1)} + 2^{(n-2)} + \cdots + 2 + 1 &=& S 
\\  2^{n} + 2^{(n-1)} + .... + 2^2 + 2 &=& 2S \tag {multiply both sides by 2}
\\  2^{n} + 2^{(n-1)} + .... + 2^2 + 2 +1 &=& 2S +1\tag {add 1 to both sides }
\\  2^{n} + S &=& 2S +1\tag {replace the term on LHS by S }
\\  2^{n}  &=& S +1\tag {subtract S from both sides }
\\  2^{n} -1  &=& S
\\  S &=& 2^{n} -1 \quad \square  
\end{eqnarray}$$
</s>