\(A=2+2^3+...+2^{101}\)
\(4A=2^3+2^5+...+2^{101}+2^{103}\)
\(4A-A=2^{103}-2\)
\(3A=2^{103}-2\)
\(A=\dfrac{2^{103}-2}{3}\)
\(\Rightarrow1+2+2^3+...+2^{101}=A+1=\dfrac{2^{103}+1}{3}\)
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