\(M=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+.....+\frac{1}{37\cdot38}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{37}-\frac{1}{38}\)
\(=\left(\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{37}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{38}\right)\)
\(=\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{38}\right)-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+....+\frac{1}{38}\right)\)
\(=\frac{1}{20}+\frac{1}{21}+\frac{1}{22}+...+\frac{1}{38}\)
\(N=\frac{1}{20\cdot38}+\frac{1}{21\cdot37}+...+\frac{1}{38\cdot20}\)
\(\Rightarrow58N=\frac{1}{20}+\frac{1}{38}+\frac{1}{21}+\frac{1}{37}+...+\frac{1}{37}+\frac{1}{20}\)
\(=2\left(\frac{1}{20}+\frac{1}{21}+\frac{1}{22}+...+\frac{1}{38}\right)\)
\(=2A\)
\(\Rightarrow N=\frac{2}{58}M\)
\(\Rightarrow\frac{M}{N}=29\)là số nguyên.
