Question

Chững tỏ rằng:

1/2+1/2 mũ 2+1/2 mũ 3+...+1/2 mũ 2012<1

Answer 1

cm \(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}< 1\)

Đặt \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\)

\(\frac{1}{2}A=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{2013}}\)

\(\frac{1}{2}A-A=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{2013}}-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)\)

\(-\frac{1}{2}A=\frac{1}{2^{2013}}-\frac{1}{2}\)

\(\Leftrightarrow A=1-\frac{1}{2^{2012}}< 1\)

\(\RightarrowĐPCM\)