Question

Chứng tỏ rằng:

1)1-\(\frac{1}{2}\)-\(\frac{1}{2^2}\)-\(\frac{1}{2^3}\)-\(\frac{1}{2^4}\)-...-\(\frac{1}{2^{10}}\)>\(\frac{1}{2^{11}}\)

Answer 1

\(1-\frac{1}{2}-\frac{1}{2^2}-...-\frac{1}{2^{10}}\)

\(=1-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)\)(1)

Đặt \(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\)

\(\Rightarrow2A=1+\frac{1}{2}+...+\frac{1}{2^9}\)

\(\Rightarrow2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^9}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)\)

\(\Rightarrow A=1-\frac{1}{2^{10}}\)

Thay A vào (1)

\(\Rightarrow1-\left(1-\frac{1}{2^{10}}\right)\)

\(=1-1+\frac{1}{2^{10}}=\frac{1}{2^{10}}\)

Ta có: 2¹⁰ < 2¹¹

\(\Rightarrow\frac{1}{2^{10}}>\frac{1}{2^{11}}\)(đpcm)