[tex]\frac{1}{1 \cdot 3}=\frac{\frac{1}{1} -\frac{1}{3} }{2} \\\frac{1}{x(x+2)}=\frac{\frac{1}{x}-\frac{1}{x+2} }{2} \\\frac{1}{1 \cdot 3}+...+\frac{1}{79 \cdot 81}=\frac{\frac{1}{1} -\frac{1}{3} }{2}+\frac{\frac{1}{3} -\frac{1}{5} }{2}+...+\frac{\frac{1}{79} -\frac{1}{81} }{2}=\frac{\frac{1}{1} -\frac{1}{81} }{2}=\frac{40}{81}[/tex]