Tính \(\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{{98}} - \frac{1}{{99}} + \frac{1}{{99}} - \frac{1}{{100}}\)
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Lời giải:
Báo sai\(\begin{array}{l} \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{{98}} - \frac{1}{{99}} + \frac{1}{{99}} - \frac{1}{{100}}\\ = \frac{1}{2} + \left( { - \frac{1}{3} + \frac{1}{3}} \right) + \left( { - \frac{1}{4} + \frac{1}{4}} \right) + \left( { - \frac{1}{5} + \frac{1}{5}} \right) + ... + \left( { - \frac{1}{{99}} + \frac{1}{{99}}} \right) - \frac{1}{{100}}\\ = \frac{1}{2} + \,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\, + ... + \,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{1}{{100}}\\ = \frac{1}{2} - \frac{1}{{100}} = \frac{{50}}{{100}} - \frac{1}{{100}} = \frac{{50 - 1}}{{100}} = \frac{{49}}{{100}} \end{array}\)