Question

A= 1\2+1\2^2+..+ 1\2^101 Tính A

Answer 1

\(A=\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{101}}\)

\(2A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{100}}\)

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

\(A=1-\frac{1}{2^{101}}\)

Answer 2

\(A=\frac{1}{2}+\frac{1}{2^2}+..+\frac{1}{2^{101}}\Rightarrow2A=1+\frac{1}{2}+...+\frac{1}{2^{100}}\)

\(\Rightarrow2A-A=\left(1+\frac{1}{2}+..+\frac{1}{2^{100}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+..+\frac{1}{2^{101}}\right)=1-\frac{1}{2^{101}}\)

\(=>A=1-\frac{1}{2^{101}}\)