<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] let $S= 1+x+x^2+.......+x^{n-1}$ where $x\ne1$
Multiply this equation by $x$ on both sides and get:
$xS= x+x^2+...........+x^{n}$ 
now subtract the second equality from the first one and you will end up with:
$(1-x)S= 1+x+x^2+.....+x^{n-1}-x-x^2-.......-x^n$ 
simplify:
$(1-x)S=1-x^n$ which implies that $S=\frac{1-x^n}{1-x} $
when $x=2$;  $S=\frac{1-2^n}{1-2}=2^n-1$
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