Question

\(\text{M=1-1/1.3-1/3.5-1/5.7-...-1/2013.2015}\)

Answer 1

\(M=1-\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{2013.2015}\right)\)

\(M=1-\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{2013.2015}\right)\)

\(M=1-\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{2013}-\frac{1}{2015}\right)\)

\(M=1-\frac{1}{2}.\left(1-\frac{1}{2015}\right)\)

bạn tự tính nốt nhé

Answer 2

\(M=1-\frac{1}{1.3}-\frac{1}{3.5}-\frac{1}{5.7}-...-\frac{1}{2013.2015}\)

\(\Leftrightarrow M=1-\left(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2013.2015}\right)\)

\(\Leftrightarrow M=1-\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{2013.2015}\right)\)

\(\Leftrightarrow M=1-\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2013}-\frac{1}{2015}\right)\)

\(\Leftrightarrow M=1-\frac{1}{2}\left(1-\frac{1}{2015}\right)\)

\(\Leftrightarrow M=1-\frac{1}{2}.\frac{2014}{2015}\)

\(\Leftrightarrow M=1-\frac{2014}{4030}\)

\(\Leftrightarrow M=\frac{2016}{4030}=\frac{1008}{2015}\)