[tex]\dfrac{2}{1\cdot 3}+\dfrac{2}{3\cdot 5}+\dfrac{2}{5\cdot 7}+...+\dfrac{2}{79\cdot 81} = \\ \\\\ =\dfrac{3-1}{1\cdot 3}+\dfrac{5-3}{3\cdot 5}+\dfrac{7-5}{5\cdot 7}+...+\dfrac{81-79}{79\cdot 81} = \\ \\\\ = \Big(\dfrac{3}{1\cdot 3}-\dfrac{1}{1\cdot 3}\Big) + \Big(\dfrac{5}{3\cdot 5}-\dfrac{3}{3\cdot 5}\Big) + ... +\Big(\dfrac{81}{79\cdot 81}-\dfrac{79}{79\cdot 81}\Big) =\\ \\ \\ =1 - \dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{79}-\dfrac{1}{81} =[/tex]
[tex]\\= 1+\dfrac{1}{3}+\dfrac{1}{5}+\dfrac{1}{7}+...+\dfrac{1}{79} - \dfrac{1}{3}-\dfrac{1}{5}-\dfrac{1}{7}-...-\dfrac{1}{79}-\dfrac{1}{81} = \\ \\\\ = 1 - \dfrac{1}{81} = \dfrac{81-1}{81}=\boxed{\dfrac{80}{81}}[/tex]
