$\begin{array}{l}
S = 1 + 2 + {2^2} + ... + {2^{100}}\\
\Rightarrow 2.S = 2.\left( {1 + 2 + {2^2} + ... + {2^{100}}} \right) = 2 + {2^2} + ... + {2^{101}}\\
\Rightarrow 2.S - S = 2 + {2^2} + ... + {2^{101}} - \left( {1 + 2 + {2^2} + ... + {2^{100}}} \right)\\
\Rightarrow S = 2 + {2^2} + ... + {2^{101}} - 1 - 2 - {2^2} - ... - {2^{100}}\\
\Rightarrow S = {2^{101}} - 1
\end{array}$
