Gọi \(d\) là công sai của cấp số đã cho

Ta có: \({S_{100}} = 50\left( {2{u_1} + 99d} \right) = 24850 \Rightarrow d = \frac{{497 - 2{u_1}}}{{99}} = 5\)

\( \Rightarrow 5S = \frac{5}{{{u_1}{u_2}}} + \frac{5}{{{u_2}{u_3}}} + ... + \frac{5}{{{u_{49}}{u_{50}}}}\)

          \( = \frac{{{u_2} - {u_1}}}{{{u_1}{u_2}}} + \frac{{{u_3} - {u_2}}}{{{u_2}{u_3}}} + ... + \frac{{{u_{50}} - {u_{49}}}}{{{u_{49}}{u_{50}}}}\)

         \( = \frac{1}{{{u_1}}} - \frac{1}{{{u_2}}} + \frac{1}{{{u_2}}} - \frac{1}{{{u_3}}} + ... + \frac{1}{{{u_{48}}}} - \frac{1}{{{u_{49}}}} + \frac{1}{{{u_{49}}}} - \frac{1}{{{u_{50}}}}\)

        \( = \frac{1}{{{u_1}}} - \frac{1}{{{u_{50}}}} = \frac{1}{{{u_1}}} - \frac{1}{{{u_1} + 49d}} = \frac{{245}}{{246}}\)

\( \Rightarrow S = \frac{{49}}{{246}}\).