\(M=\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
\(\Rightarrow M=\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{49.51}\)
\(\Rightarrow2M=\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{49.51}\)
\(\Rightarrow2M=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\)
\(\Rightarrow2M=\frac{1}{3}-\frac{1}{51}\)
\(\Rightarrow2M=\frac{16}{51}\)
\(\Rightarrow M=\frac{8}{51}\)
\(N=\frac{-5}{1.3}+\frac{-5}{3.5}+...+\frac{-5}{2013.2015}\)
\(\Rightarrow N=-\frac{5}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{2013.2015}\right)\)
\(\Rightarrow N=-\frac{5}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{2013}-\frac{1}{2015}\right)\)
\(\Rightarrow N=-\frac{5}{2}\left(1-\frac{1}{2015}\right)\)
\(\Rightarrow N=-\frac{5}{2}.\frac{2014}{2015}\)
\(\Rightarrow N=-\frac{1007}{403}\)
