Tính giá trị biểu thức

Question

1/2-1/2^2+1/2^3+1/2^4+…+1/2^99-1/2^100 tính giá trị biểu thức

Answer 1

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

Ta có: \(2A=1-\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+...+\frac{1}{2^{98}}-\frac{1}{2^{99}}\)

\(\Rightarrow A+2A=(\frac{1}{2}-\frac{1}{2^{2}}+\frac{1}{2^{3}}-\frac{1}{2^{4}}+...+\frac{1}{2^{99}}-\frac{1}{2^{100}})+(1-\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+...+\frac{1}{2^{98}}-\frac{1}{2^{99}})\)

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

\(\Leftrightarrow 3A=1-\frac{1}{2^{100}}\)

Vậy

\(A=\frac{1}3{}-\frac{1}{3.2^{100}}=\frac{2^{100}-1}{3.2^{100}}\)