[tex]\displaystyle\\
\left(1+ \frac{1 }{2}\right)\cdot \left(1+ \frac{1 }{3}\right)\cdot \left(1+ \frac{1 }{4}\right)\cdot ...\cdot \left(1+ \frac{1 }{99}\right)\cdot \left(1+ \frac{1 }{100}\right)=\\\\\\
=\left(\frac{2}{2}+\frac{1 }{2}\right)\cdot \left(\frac{3}{3}+ \frac{1 }{3}\right)\cdot \left(\frac{4}{4}+ \frac{1 }{4}\right)\cdot ...\cdot \left(\frac{99}{99}+ \frac{1 }{99}\right)\cdot \left(\frac{100}{100}+ \frac{1 }{100}\right)=
[/tex]
[tex]\displaystyle\\
=\frac{3 }{2}\cdot \frac{4 }{3}\cdot \frac{5 }{4}\cdot ...\cdot \frac{100}{99} \cdot \frac{101}{100}=\\\\
\text{(Se simplifica numaratorul de la fractia precedenta cu numitorul}\\
\text{de la fractia urmatoare.)}\\\\
=\frac{1 }{2}\cdot \frac{101}{1}= \boxed{\bf \frac{101}{2}}
[/tex]
