Hello,
[tex] \dfrac{1}{n} - \dfrac{1}{n+1} = \dfrac{1}{n(n+1)} \\
\dfrac{1}{1*2} = \dfrac{1}{1} - \dfrac{1}{2} \\
\dfrac{1}{2*3} = \dfrac{1}{2} - \dfrac{1}{3} \\
\dfrac{1}{3*4} = \dfrac{1}{3} - \frac{1}{4} \\
...\\
\dfrac{1}{n*(n+1)} = \dfrac{1}{n} - \dfrac{1}{n+1} \\
[/tex]
Adding member by member, we have
[tex] \dfrac{1}{1*2} + \dfrac{1}{2*3} +\dfrac{1}{3*4} +...\dfrac{1}{n*(n+1)}=\\
\dfrac{1}{1} - \dfrac{1}{n+1} \\
= \dfrac{n}{n+1} \\
[/tex]
if n=100 sum [tex]\boxed{= \dfrac{100}{101} }[/tex]
