Giải thích các bước giải:

$\begin{aligned}
& \text{Đặt}: \\
& A = 1 - 2 + 2^2 - 2^3 + 2^4 - 2^5 + \dots - 2^{99} + 2^{100} \\
& B = (-1) \cdot (1 + 2^2 + 2^4 + \dots + 2^{98}) + 2^{100} \\
& \\
& \text{Ta có:} \\
& A = 1 - 2 + 2^2 - 2^3 + \dots - 2^{99} + 2^{100} \\
& 2A = 2 - 2^2 + 2^3 - 2^4 + \dots - 2^{100} + 2^{101} \\
& A + 2A = (1 - 2 + \dots + 2^{100}) + (2 - 2^2 + \dots + 2^{101}) \\
& 3A = 1 + (-2+2) + (2^2-2^2) + \dots + (-2^{100}+2^{100}) + 2^{101} \\
& 3A = 1 + 2^{101} \\
& A = \dfrac{2^{101} + 1}{3} \\
& \\
& \text{Ta có:} \\
& B = -(1 + 2^2 + 2^4 + \dots + 2^{98}) + 2^{100} \\
& B = - \left( \dfrac{(2^2)^{50} - 1}{2^2 - 1} \right) + 2^{100} \\
& B = - \dfrac{2^{100} - 1}{3} + \dfrac{3 \cdot 2^{100}}{3} \\
& B = \dfrac{-(2^{100} - 1) + 3 \cdot 2^{100}}{3} \\
& B = \dfrac{-2^{100} + 1 + 3 \cdot 2^{100}}{3} \\
& B = \dfrac{2 \cdot 2^{100} + 1}{3} \\
& B = \dfrac{2^{101} + 1}{3} \\
& \\
& \left\{ \begin{aligned} & A = \dfrac{2^{101} + 1}{3} \\ & B = \dfrac{2^{101} + 1}{3} \end{aligned} \right. \\
& \Rightarrow A = B
\end{aligned}$