Question

[Nguyen Thuy]

A=1+3/2^3+4/2^4+5/2^5+...+100/2^100

Answer 1

[Đình Sang Bùi]

\(\frac{1}{2}.A=\frac{1}{2}+\frac{3}{2^4}+\frac{4}{2^5}+\frac{5}{2^6}+...+\frac{100}{2^{101}}\)

\(A-\frac{1}{2}.A=\frac{1}{2}+\frac{3}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}+...+\frac{1}{2^{100}}-\frac{1}{2^{101}}\)

                \(=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{100}}-\frac{1}{2^{101}}\)(Do 3/2^3=1/2^2+1/2^3)

Đặt \(B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\)

  \(2B=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}\)

   \(2B-B=1-\frac{1}{2^{100}}\)

Do đó:\(A-\frac{1}{2}A=1-\frac{1}{2^{100}}-\frac{1}{2^{101}}\)

                     \(\Rightarrow A=\frac{1}{2}-\frac{1}{2^{99}}-\frac{1}{2^{100}}\)

                          \(=\frac{2^{99}-2-1}{2^{100}}\)

                        \(=\frac{2^{99}-3}{2^{100}}\)

Answer 2

[Đình Sang Bùi]

Í nhầm cho mình sửa chỗ 1/2^101 thành 100/2^101 nhé

Mấy chỗ khác bạn từ đó sửa lại nha

Sorry