Question

B=1/5+1/5^2+1/5^3+...+1/5^2014.Chứng minh B<1/4

Answer 1

\(B=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}\)

\(5B=5\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}\right)\)

\(5B=1+\frac{1}{5}+...+\frac{1}{5^{2013}}\)

\(5B-B=\left(1+\frac{1}{5}+...+\frac{1}{5^{2013}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}\right)\)

\(4B=1-\frac{1}{5^{2014}}\Rightarrow B=\frac{1-\frac{1}{5^{2014}}}{4}\)

Ta có: \(1-\frac{1}{5^{2014}}< 1\Rightarrow\frac{1-\frac{1}{5^{2014}}}{4}< \frac{1}{4}\)

\(\Rightarrow B< \frac{1}{4}\)(Đpcm)