\[\begin{array}{l}\frac{1}{{1 \cdot 7}} = \frac{1}{6} \cdot \left( {\frac{1}{1} - \frac{1}{7}} \right)\\\frac{1}{{7 \cdot 13}} = \frac{1}{6} \cdot \left( {\frac{1}{7} - \frac{1}{{13}}} \right)\end{array}\]

\[\begin{array}{l}\frac{1}{{1 \cdot 7}} = \frac{1}{6} \cdot \left( {\frac{1}{1} - \frac{1}{7}} \right)\\\frac{1}{{7 \cdot 13}} = \frac{1}{6} \cdot \left( {\frac{1}{7} - \frac{1}{{13}}} \right)\\...\\{\text{Поэтому}}\\S = \frac{1}{{1 \cdot 7}} + \frac{1}{{7 \cdot 13}} + \frac{1}{{13 \cdot 19}} + \frac{1}{{19 \cdot 25}} + ... + \frac{1}{{91 \cdot 97}} = \\\frac{1}{6}\left( {1 - \frac{1}{7} + \frac{1}{7} - \frac{1}{{13}} + \frac{1}{{13}} - ... - \frac{1}{{91}} + \frac{1}{{91}} - \frac{1}{{97}}} \right) = \\\frac{1}{6} \cdot \left( {1 - \frac{1}{{97}}} \right) = \frac{1}{6} \cdot \frac{{96}}{{97}} = \frac{{16}}{{97}}\end{array}\]

\[\frac{{16}}{{97}}\]