Question

cho A = 1/3+1/3^2+1/3^3+...+1/3^2016 

chứng tỏ A < 1/2

Answer 1

\(\frac{1}{3}A=\left(\frac{1}{3}\right)^2+\left(\frac{1}{3}\right)^3+...+\left(\frac{1}{3}\right)^{2017}\)

\(A-\frac{1}{3}A=\frac{1}{3}-\left(\frac{1}{3}\right)^{2017}\)

\(A=\frac{2}{3}\left[\frac{1}{3}-\left(\frac{1}{3}\right)^{2017}\right]\)

\(A=\frac{2}{9}-\frac{2}{3}.\left(\frac{1}{3}\right)^{2017}\)

\(\frac{2}{9}< \frac{1}{2};\frac{2}{3}.\left(\frac{1}{3}\right)^{2017}>0\Rightarrow A< \frac{1}{2}\)