Question

2/(2 x 4) + 2/(4 x 6) + 2/(6 x 8) + ..... + 2/(2014 x 2016)

Answer 1

\(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+....+\frac{2}{2014.2016}\)

\(=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+......+\frac{1}{2014}-\frac{1}{2016}\)

\(=\frac{1}{2}-\frac{1}{2016}\)

\(=\frac{1008}{2016}-\frac{1}{2016}=\frac{1007}{2016}\)

Answer 2

\(\frac{2}{2\times4}+\frac{2}{4\times6}+...+\frac{2}{2014\times2016}\)

=\(\left(\frac{2}{2}-\frac{2}{4}\right)+\left(\frac{2}{4}-\frac{2}{6}\right)+...+\left(\frac{2}{2014}-\frac{2}{2016}\right)\)

=\(\frac{2}{2}-\frac{2}{4}+\frac{2}{4}-\frac{2}{6}+...+\frac{2}{2014}-\frac{2}{2016}\)

=\(\frac{2}{2}-\frac{2}{2016}=\frac{1007}{1008}\)