Tính biểu thức 7/ 1.3 + 7/3.5 + 7/5.7 + ... + 7/99.101

\(\frac{7}{{1 \cdot 3}} + \frac{7}{{3 \cdot 5}} + \frac{7}{{5 \cdot 7}} + ... + \frac{7}{{99 \cdot 101}}\)

\( = \frac{7}{2}\left( {\frac{2}{{1 \cdot 3}} + \frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + ... + \frac{2}{{99 \cdot 101}}} \right)\)

\( = \frac{7}{2}\left( {1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + ... + \frac{1}{{99}} - \frac{1}{{101}}} \right)\)

\( = \frac{7}{2}\left( {1 - \frac{1}{{101}}} \right)\)

\( = \frac{7}{2} \cdot \frac{{100}}{{101}}\)

\( = \frac{{350}}{{101}}\).