\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+.....+\frac{1}{2016\cdot2017}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{2016}-\frac{1}{2017}\)
\(=1-\frac{1}{2017}=\frac{2016}{2017}\)
= 1/1-1/2+1/2-1/3+1/3-............-1/2017
=1-1/2017
=2016/2017
\(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+\(\frac{1}{3.4}\)+..+\(\frac{1}{2006.2007}\)
=1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+..+\(\frac{1}{2006}\)-\(\frac{1}{2007}\)
=1-\(\frac{1}{2007}\)=\(\frac{2006}{2007}\)
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2016.2017}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{2016}-\frac{1}{2017}\)
\(=1-\frac{1}{2017}\)
\(=\frac{2017}{2017}-\frac{1}{2017}\)
\(=\frac{2016}{2017}\)
\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2016\cdot2017}\)
\(\Rightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2016}-\frac{1}{2017}\)
\(\Rightarrow1-\frac{1}{2017}\)
\(\Rightarrow\frac{2016}{2017}\)
1/(1.2) + 1/(2.3) + 1/(3.4) + ......... + 1/(2016.2017)
= 1 - 1/2 + 1/2 - 1/2 + 1/3 - 1/4 + ......... + 1/2016 - 1/2017
= 1 - 1/2017
= 2016/2017
