1/2 x 4 + 1/4 x 6 + 1/6 x 8 + .... 1/2014 x 2016
= 1/2 - 1/4 + 1/4 - 1/6 + 1/6 - 1/8 + .... + 1/2014 - 1/2016
= 1/2 - 1/2016
= 1007/2016
\(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2014.2016}\)
\(=\frac{1.2}{2.4.2}+\frac{1.2}{4.6.2}+\frac{1.2}{6.8.2}+......+\frac{1.2}{2014.2016.2}\)
\(=\frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+.....+\frac{2}{2014.2016}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+.....+\frac{1}{2014}-\frac{1}{2016}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2016}\right)\)
\(=\frac{1}{2}.\frac{1007}{2016}\)
\(=\frac{1007}{4032}\)
