Question

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

Answer 1

Ta có:

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

\(\Rightarrow\dfrac{1}{2}B=\dfrac{1}{2^2}-\dfrac{1}{2^3}+\dfrac{1}{2^4}-....+\dfrac{1}{2^{100}}-\dfrac{1}{2^{101}}\)

\(\Rightarrow B+\dfrac{1}{2}B=\left(\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-...+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}\right)+\left(\dfrac{1}{2^2}-\dfrac{1}{2^3}+\dfrac{1}{2^4}-...+\dfrac{1}{2^{100}}-\dfrac{1}{2^{101}}\right)\)

\(\Rightarrow\dfrac{3}{2}B=\dfrac{1}{2}-\dfrac{1}{2^{101}}\)

\(\Rightarrow B=\dfrac{\dfrac{1}{2}-\dfrac{1}{2^{101}}}{\dfrac{3}{2}}\)