Giải thích các bước giải:
Áp dụng:
\[\frac{1}{{n\left( {n + 1} \right)}} = \frac{{\left( {n + 1} \right) - n}}{{n\left( {n + 1} \right)}} = \frac{1}{n} - \frac{1}{{n + 1}}\]
Ta có:
\[\begin{array}{l}
P = \frac{1}{{100.99}} - \frac{1}{{99.98}} - \frac{1}{{98.97}} - \frac{1}{{97.96}} - ..... - \frac{1}{{2.1}}\\
\Leftrightarrow P = \frac{1}{{100.99}} - \left( {\frac{1}{{99.98}} + \frac{1}{{98.97}} + \frac{1}{{97.96}} + ... + \frac{1}{{2.1}}} \right)\\
\Leftrightarrow P = \frac{1}{{99}} - \frac{1}{{100}} - \left( {\frac{1}{{98}} - \frac{1}{{99}} + \frac{1}{{97}} - \frac{1}{{98}} + \frac{1}{{96}} - \frac{1}{{97}} + ... + 1 - \frac{1}{2}} \right)\\
\Leftrightarrow P = \frac{1}{{99}} - \frac{1}{{100}} - \left( {1 - \frac{1}{{99}}} \right)\\
\Rightarrow P = \frac{1}{{99}} - \frac{1}{{100}} - 1 + \frac{1}{{99}} = \frac{2}{{99}} - \frac{{101}}{{100}}
\end{array}\]
