[tex]\displaystyle\\\text{Folosim formula: }~\frac{7}{n(n+7)}=\frac{1}{n}-\frac{1}{n+7}\\\\ \text{Rezolvare:}\\\\a)\\\\\frac{7}{1\cdot8}+\frac{7}{8\cdot15}+\frac{7}{15\cdot22}+...+\frac{7}{(7n-6)(7n+1)}-\frac{1}{7n+1}=\\\\=\frac{1}{1}-\frac{1}{8}+\frac{1}{8}-\frac{1}{15}+\frac{1}{15}-\frac{1}{22}+...+\frac{1}{(7n-6)}-\frac{1}{(7n+1)}+\frac{1}{7n+1}=\\\\\text{(Se reduc termenii asemenea si ramanem cu:)}\\\\=\frac{1}{1}-\frac{1}{7n+1}+\frac{1}{7n+1}=1[/tex]
