[tex]S=\frac{1}{1\cdot5}+\frac{1}{5\cdot9}+\frac{1}{9\cdot13}+\ldots+\frac{1}{81\cdot85}=\\\\=\frac{1}{4}\cdot\left(\frac{1}{1}-\frac{1}{5}\right)+\frac{1}{4}\cdot\left(\frac{1}{5}-\frac{1}{9}\right)+\frac{1}{4}\cdot\left(\frac{1}{9}-\frac{1}{13}\right)+\ldots+\frac{1}{4}\cdot\left(\frac{1}{81}-\frac{1}{85}\right)=\\\\=\frac{1}{4}\cdot\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\ldots+\frac{1}{77}-\frac{1}{81}+\frac{1}{81}-\frac{1}{85}\right)=\\\\=\frac{1}{4}\cdot\left(1-\frac{1}{85}\right)=\frac{1}{4}\cdot\frac{84}{85}=\frac{21}{85}.[/tex]
