Tính nhanh: \(B = \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}\).

Lời giải

Ta có \(B = \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}\)

\( = \frac{1}{2}\left( {\frac{2}{{3.5}} + \frac{2}{{5.7}} + \frac{2}{{7.9}} + \frac{2}{{9.11}} + \frac{2}{{11.13}}} \right)\)

\( = \frac{1}{2}\left( {\frac{{5 - 3}}{{3.5}} + \frac{{7 - 5}}{{5.7}} + \frac{{9 - 7}}{{7.9}} + \frac{{11 - 9}}{{9.11}} + \frac{{13 - 11}}{{11.13}}} \right)\)

\[ = \frac{1}{2}\left( {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{13}}} \right)\]

\[ = \frac{1}{2}\left( {\frac{1}{3} - \frac{1}{{13}}} \right)\]\[ = \frac{1}{2}.\frac{{10}}{{39}} = \frac{5}{{39}}\].