Tính giá trị của biểu thức: \(H = \frac{{38}}{{50}} + \frac{9}{{20}} - \frac{{11}}{{30}} + \frac{{13}}{{42}} - \frac{{15}}{{56}} + \frac{{17}}{{72}} - ... + \frac{{197}}{{9702}} - \frac{{199}}{{9900}}\)

Ta có:

\(\begin{array}{l} \frac{9}{{20}} = \frac{9}{{4.5}} = \frac{{5 + 4}}{{4.5}} = \frac{5}{{4.5}} + \frac{4}{{4.5}} = \frac{1}{4} + \frac{1}{5}\\ \frac{{11}}{{30}} = \frac{{11}}{{5.6}} = \frac{{6 + 5}}{{5.6}} = \frac{6}{{5.6}} + \frac{5}{{5.6}} = \frac{1}{5} + \frac{1}{6}\\ \frac{{13}}{{42}} = \frac{{13}}{{6.7}} = \frac{{7 + 6}}{{6.7}} = \frac{7}{{6.7}} + \frac{6}{{6.7}} = \frac{1}{6} + \frac{1}{7}\\ ...\\ \frac{{197}}{{9702}} = \frac{{197}}{{98.99}} = \frac{{99 + 98}}{{98.99}} = \frac{{99}}{{98.99}} + \frac{{98}}{{98.99}} = \frac{1}{{98}} + \frac{1}{{99}}\\ \frac{{199}}{{9900}} = \frac{{199}}{{99.100}} = \frac{{100 + 99}}{{99.100}} = \frac{{100}}{{99.100}} + \frac{{99}}{{99.100}} = \frac{1}{{99}} + \frac{1}{{100}} \end{array}\)

Do đó 

\(\begin{array}{l} H = \frac{{38}}{{50}} + \frac{9}{{20}} - \frac{{11}}{{30}} + \frac{{13}}{{42}} - \frac{{15}}{{56}} + \frac{{17}}{{72}} - ... + \frac{{197}}{{9702}} - \frac{{199}}{{9900}}\\ = \frac{{38}}{{50}} + \left( {\frac{1}{4} + \frac{1}{5}} \right) - \left( {\frac{1}{5} + \frac{1}{6}} \right) + \left( {\frac{1}{6} + \frac{1}{7}} \right) - \left( {\frac{1}{7} + \frac{1}{8}} \right)\\ + \left( {\frac{1}{8} + \frac{1}{9}} \right) - ... + \left( {\frac{1}{{98}} + \frac{1}{{99}}} \right) - \left( {\frac{1}{{99}} + \frac{1}{{100}}} \right)\\ = \frac{{38}}{{50}} + \frac{1}{4} + \frac{1}{5} - \frac{1}{5} - \frac{1}{6} + \frac{1}{6} + \frac{1}{7} - \frac{1}{7} - \frac{1}{8} + \frac{1}{8} + \frac{1}{9}\\ - ... + \frac{1}{{98}} + \frac{1}{{99}} - \frac{1}{{99}} - \frac{1}{{100}}\\ = \frac{{38}}{{50}} + \frac{1}{4} + \left( {\frac{1}{5} - \frac{1}{5}} \right) + \left( { - \frac{1}{6} + \frac{1}{6}} \right) + \left( {\frac{1}{7} - \frac{1}{7}} \right) + \left( { - \frac{1}{8} + \frac{1}{8}} \right)\\ + \left( {\frac{1}{9} - \frac{1}{9}} \right) + ... + \left( { - \frac{1}{{98}} + \frac{1}{{98}}} \right) + \left( {\frac{1}{{99}} - \frac{1}{{99}}} \right) - \frac{1}{{100}}\\ = \frac{{38}}{{50}} + \frac{1}{4} + 0 + 0 + 0 + 0 + 0 + ... + 0 + 0 - \frac{1}{{100}}\\ = \frac{{38}}{{50}} + \frac{1}{4} - \frac{1}{{100}}\\ = \frac{{76}}{{100}} + \frac{{25}}{{100}} - \frac{1}{{100}}\\ = \frac{{76 + 25 - 1}}{{100}}\\ = \frac{{100}}{{100}} = 1 \end{array}\)

Vậy H = 1.