2 + 2^2 + 2^3 + \ldots + 2^n = 2^1 + 2^2 + 2^3 + \ldots + 2^n = 2 (2^0 + 2^1 + 2^2 + 2^3 + \ldots + 2^{n-1}) = 2 (1 + 2^1 + 2^2 + 2^3 + \ldots + 2^{n-1}) = 2 (2^{n-1} - 1) = 2^{n+1}-2