Question 1183173
This is an alternative solution.

It is easy to see by direct multiplication that 


(1-x)*(1 + x + x^2 + x^3 +...+ x^(n-1)) = 1 - x^n.
  

If you let x = 2 in the above equation, you will get 


(-1)*(1 + 2 + 2^2 + 2^3 +...+ 2^(n-1)) = 1-2^n


This implies then that 


1 + 2 + 2^2 + 2^3 +...+ 2^(n-1) = 2^n -1,


after multiplying both sides of the equation by -1.