Question

Cho A=\(\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+.....+\left(\frac{1}{2}\right)^{2014}\)

Chứng minh A<1

Answer 1

\(A=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{2014}\)

\(A=\frac{1}{2}+\frac{1^2}{2^2}+...+\frac{1^{2014}}{2^{2014}}\)

\(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\)

\(2A=2\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\right)\)

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

\(2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^{2013}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\right)\)

\(A=1-\frac{1}{2^{2014}}< 1\)

Đpcm