Question

\(S=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

Answer 1

\(S=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(2S=2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\)

\(2S+S=\left(2^{101}-2^{100}+2^{99}-...-2^2\right)-\left(2^{100}-2^{99}+2^{98}-...-2\right)\)

\(3S=2^{101}-2\)

\(S=\frac{2^{101}-2}{3}\)

Answer 2

\(S=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(\Rightarrow\) \(2S=2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\)

\(\Rightarrow\) \(3S=2^{101}-2\)

\(\Rightarrow\) \(S=\frac{2^{101}-2}{3}\)