Phương pháp:
- Tìm CSC đã cho bằng cách sử dụng công thức \[{S_n} = \frac{{n\left[ {2{u_1} + \left( {n - 1} \right)d} \right]}}{2}\]
- Thay vào tổng đã cho tính toán.
Cách giải:
Ta có: \[24850 = {S_{100}} = \frac{{100\left( {2.1 + 99d} \right)}}{2} \Leftrightarrow d = 5\]
Khi đó \[{u_1} = 1,{\rm{ }}{u_2} = 6,{\rm{ }}{u_3} = 11,{\rm{ }}{u_4} = 16,...,{u_{49}} = {u_1} + 48d = 241,{\rm{ }}{u_{50}} = {u_1} + 49d = 246\]
\[ \Rightarrow S = \frac{1}{{{u_1}{u_2}}} + \frac{1}{{{u_2}{u_3}}} + ... + \frac{1}{{{u_{49}}{u_{50}}}} = \frac{1}{{1.6}} + \frac{1}{{6.11}} + \frac{1}{{11.16}} + ... + \frac{1}{{241.246}}\] \[ = \frac{1}{5}\left( {\frac{1}{1} - \frac{1}{6}} \right) + \frac{1}{5}\left( {\frac{1}{6} - \frac{1}{{11}}} \right) + ... + \frac{1}{5}\left( {\frac{1}{{241}} - \frac{1}{{246}}} \right)\]
\[ = \frac{1}{5}\left( {1 - \frac{1}{6} + \frac{1}{6} - \frac{1}{{11}} + ... + \frac{1}{{241}} - \frac{1}{{246}}} \right)\]
\[ = \frac{1}{5}\left( {1 - \frac{1}{{246}}} \right) = \frac{{49}}{{246}}\]
Vậy \[S = \frac{{49}}{{246}}\].