[tex]\displaystyle\\
\texttt{Folosim formula: }~~\frac{1}{n\times (n+1)}=\frac{1}{n}-\frac{1}{ (n+1)}\\\\\\
\frac{1}{1\times 2}+\frac{1}{2\times 3}+ \frac{1}{3\times 4}+\cdots + \frac{1}{50\times 51}+\frac{1}{51\times 52} =\\\\
= \frac{1}{1} - \frac{1}{2} +\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots \frac{1}{50}-\frac{1}{51}+\frac{1}{51}-\frac{1}{52} =\\\\
\texttt{Termenii se reduc cate doi in afara de primul si ultimul.}\\\\
=\frac{1}{1} -\frac{1}{52} = \boxed{\frac{51}{52} }[/tex]
