Question

Rút gọn: M=\(2^{100}-2^{99}+2^{98}-2^{97}+....+2^2-2\)

Answer 1

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

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

=> 2M+M=3M\(\left(2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\right)+\left(2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\right)\)=\(2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2+2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

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

= \(2^{101}-2\)

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