Tính giá trị của biểu thức: \(M = + + + \ldots +

* Hướng dẫn giải

\(M = \frac{{{3^2}}}{{2.5}} + \frac{{{3^2}}}{{5.8}} + \frac{{{3^2}}}{{8.11}} +  \ldots  + \frac{{{3^2}}}{{98.101}}\)

\(\begin{array}{l}M = \frac{{{3^2}}}{{2.5}} + \frac{{{3^2}}}{{5.8}} + \frac{{{3^2}}}{{8.11}} +  \ldots  + \frac{{{3^2}}}{{98.101}}\\\,\,\,\,\,\,\, = 3.\left( {\frac{3}{{2.5}} + \frac{3}{{5.8}} + \frac{3}{{8.11}} +  \ldots  + \frac{3}{{98.101}}} \right)\\\,\,\,\,\,\,\, = 3.\left( {\frac{1}{2} - \frac{1}{5} + \frac{1}{5} - \frac{1}{8} + \frac{1}{8} - \frac{1}{{11}} +  \ldots  + \frac{1}{{98}} - \frac{1}{{101}}} \right)\\\,\,\,\,\,\,\, = 3.\left( {\frac{1}{2} - \frac{1}{{101}}} \right)\\\,\,\,\,\,\,\, = 3 \cdot \frac{{99}}{{202}}\\\,\,\,\,\,\,\, = \frac{{297}}{{202}}\end{array}\)

Vậy \(M = \frac{{297}}{{202}}\).