\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+.....+\frac{1}{36.37.38}+\frac{1}{37.38.39}\)
\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{36.37}-\frac{1}{37.38}+\frac{1}{37.38}-\frac{1}{38.39}\right)\)
\(A=\frac{1}{2}.\left(\frac{1}{2}+\left(\frac{1}{2.3}-\frac{1}{2.3}\right)+\left(\frac{1}{3.4}-\frac{1}{3.4}\right)+.....+\left(\frac{1}{37.38}-\frac{1}{37.38}\right)-\frac{1}{38.39}\right)\)
\(A=\frac{1}{2}.\left(\frac{1}{2}+0+0+.....+0-\frac{1}{38.39}\right)\)
\(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{1482}\right)\)
\(A=\frac{1}{2}.\left(\frac{741}{1482}-\frac{1}{1482}\right)\)
\(A=\frac{1}{2}.\frac{370}{1482}\)
\(A=\frac{185}{741}\)
