Tìm \(x \in \mathbb{Z},\) biết:

j) \(5 \cdot \frac{1}{6} + \frac{1}{6} \le x \le \frac{1}{4}:\frac{1}{{13}} + \frac{7}{4}\)

g) \(\frac{4}{{15}} + \frac{4}{{35}} + \frac{4}{{63}} + ... + \frac{4}{{399}}\)

\[ = \frac{4}{{3 \cdot 5}} + \frac{4}{{5 \cdot 7}} + \frac{4}{{7 \cdot 9}} + ... + \frac{4}{{19 \cdot 21}}\]

\[ = 2 \cdot \left( {\frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + \frac{2}{{7 \cdot 9}} + ... + \frac{2}{{19 \cdot 21}}} \right)\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + ... + \frac{1}{{19}} - \frac{1}{{21}}} \right)\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{{21}}} \right)\]\[ = 2 \cdot \left( {\frac{7}{{21}} - \frac{1}{{21}}} \right)\]               

\[ = 2 \cdot \frac{6}{{21}} = 2 \cdot \frac{2}{7} = \frac{4}{7}.\]

h) \[\frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}} + \frac{1}{{195}}\]

\( = \frac{1}{{1 \cdot 3}} + \frac{1}{{3 \cdot 5}} + \frac{1}{{5 \cdot 7}} + \frac{1}{{7 \cdot 9}} + \frac{1}{{9 \cdot 11}} + \frac{1}{{11 \cdot 13}} + \frac{1}{{13 \cdot 15}}\)

\( = \frac{1}{2} \cdot \left( {\frac{2}{{1 \cdot 3}} + \frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + \frac{2}{{7 \cdot 9}} + \frac{2}{{9 \cdot 11}} + \frac{2}{{11 \cdot 13}} + \frac{2}{{13 \cdot 15}}} \right)\)

\[ = \frac{1}{2} \cdot \left( {\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{13}} + \frac{1}{{13}} - \frac{1}{{15}}} \right)\]

\[ = \frac{1}{2} \cdot \left( {\frac{1}{1} - \frac{1}{{15}}} \right) = \frac{1}{2} \cdot \left( {\frac{{15}}{{15}} - \frac{1}{{15}}} \right)\]

\[ = \frac{1}{2} \cdot \frac{{14}}{{15}} = \frac{7}{{15}}.\]