Question

Cho A=\(\frac{1}{2}\)+(\(\frac{1}{2}\))\(^2\)+(\(\frac{1}{2}\))\(^3\)+...+(\(\frac{1}{2}\))\(^{2016}\)

Chứng minh:A<1

Answer 1

\(A=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+....+\left(\frac{1}{2}\right)^{2016}\)

\(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{2016}}\)

\(2A=2+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2015}}\)

\(2A-A=2+\frac{1}{2}+\frac{1}{2^2}+..+\frac{1}{2^{2015}}-\frac{1}{2}-\frac{1}{2^2}-..-\frac{1}{2^{2016}}\)

\(A=2-\frac{1}{2^{2016}}\)

\(A=\frac{2^{2009}}{2^{2010}}-\frac{1}{2^{2010}}=\frac{2^{2009}-1}{2^{2010}}\)

\(\Rightarrow A< 1\)