Đáp án:
Giải thích các bước giải:
\[\begin{array}{l}
\frac{7}{8} + \frac{{23}}{{24}} + \frac{{47}}{{48}} + \frac{{79}}{{80}} + ...... + \frac{{9799}}{{9800}} + \frac{{10199}}{{10200}}\\
= 1 - \frac{1}{8} + 1 - \frac{1}{{24}} + 1 - \frac{1}{{48}} + 1 - \frac{1}{{80}} + ..... + 1 - \frac{1}{{10200}}\\
= 1 - \frac{1}{{2 \times 4}} + 1 - \frac{1}{{4 \times 6}} + 1 - \frac{1}{{6 \times 8}} + ...... + 1 - \frac{1}{{100 \times 102}}\\
= 50 - \left( {\frac{1}{{2 \times 4}} + \frac{1}{{4 \times 6}} + \frac{1}{{6 \times 8}} + ..... + \frac{1}{{100 \times 102}}} \right)\\
= 50 - \frac{1}{{2 \times 2}} \times \left( {\frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + ..... + \frac{1}{{50 \times 51}}} \right)\\
= 50 - \frac{1}{4} \times \left( {1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + .... + \frac{1}{{50}} - \frac{1}{{51}}} \right)\\
= 50 - \frac{1}{4} \times \left( {1 - \frac{1}{{51}}} \right)\\
= 50 - \frac{1}{4} \times \frac{{50}}{{51}}\\
= 50 \times \left( {1 - \frac{1}{{4 \times 51}}} \right)\\
= 50 \times \frac{{203}}{{204}}
\end{array}\]
