Tính tổng:

\[A = \frac{{38}}{{25}} + \frac{9}{{10}} - \frac{{11}}{{15}} + \frac{{13}}{{21}} - \frac{{15}}{{28}} + \frac{{17}}{{36}} - ... + \frac{{197}}{{4851}} - \frac{{199}}{{4950}}\]

\[ = \frac{{38}}{{25}} + \frac{{18}}{{20}} - \frac{{22}}{{30}} + \frac{{26}}{{42}} - ... + \frac{{394}}{{9702}} - \frac{{398}}{{9900}}\]

\[ = \frac{{38}}{{25}} + 2\left( {\frac{9}{{20}} - \frac{{11}}{{30}} + \frac{{13}}{{42}} - ... + \frac{{197}}{{9702}} - \frac{{199}}{{9900}}} \right)\]

\[ = \frac{{38}}{{25}} + 2\left( {\frac{9}{{4 \cdot 5}} - \frac{{11}}{{5 \cdot 6}} + \frac{{13}}{{6 \cdot 7}} - ... + \frac{{197}}{{98 \cdot 99}} - \frac{{199}}{{99 \cdot 100}}} \right)\]

\[ = \frac{{38}}{{25}} + 2\left[ {\left( {\frac{1}{4} + \frac{1}{5}} \right) - \left( {\frac{1}{5} + \frac{1}{6}} \right) + \left( {\frac{1}{6} + \frac{1}{7}} \right) - ... + \left( {\frac{1}{{98}} + \frac{1}{{99}}} \right) - \left( {\frac{1}{{99}} + \frac{1}{{100}}} \right)} \right]\]

\[ = \frac{{38}}{{25}} + 2 \cdot \left( {\frac{1}{4} - \frac{1}{{100}}} \right)\]

\[ = \frac{{38}}{{25}} + 2 \cdot \left( {\frac{{25}}{{100}} - \frac{1}{{100}}} \right)\]

\[ = \frac{{38}}{{25}} + 2 \cdot \frac{{24}}{{100}}\]

\[ = \frac{{38}}{{25}} + 2 \cdot \frac{6}{{25}}\]

\[ = \frac{{38}}{{25}} + \frac{{12}}{{25}}\]

\[ = \frac{{50}}{{25}} = 2\]

Vậy A = 2.