Question

\(\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-\dfrac{1}{2^4}+...+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}\)

Answer 1

\(A=\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-\dfrac{1}{2^4}+...+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}\)

\(\Rightarrow2^{100}.A=2^{99}-2^{98}+2^{97}-2^{96}+...+2-1\)

\(\Rightarrow2^{101}.A=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(\Rightarrow2^{100}.A+2^{101}.A=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2+\left(2^{99}-2^{98}+2^{97}-2^{96}+...+2-1\right)\)

\(\Rightarrow A\left(2^{100}+2^{101}\right)=2^{100}-1\)

\(\Rightarrow A=\dfrac{2^{100}-1}{2^{100}+2^{101}}\)

Answer 2

= 1 phần 2 bạn nhé

Answer 3

1/2