chứng minh 1/1*2+1/3*4+…+1/99*100=1/51+1/52+…+1/100

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1. \(\begin{array}{l}
   \frac{1}{{1.2}} + \frac{1}{{3.4}} + ….. + \frac{1}{{99.100}} = \frac{1}{{51}} + \frac{1}{{52}} + ….. + \frac{1}{{100}}\\
   VT = \frac{1}{{1.2}} + \frac{1}{{3.4}} + ….. + \frac{1}{{99.100}}\\
    = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + …. + \frac{1}{{99}} – \frac{1}{{100}}\\
    = \left( {1 + \frac{1}{3} + \frac{1}{5} + ….. + \frac{1}{{99}}} \right) – \left( {\frac{1}{2} + \frac{1}{4} + …. + \frac{1}{{100}}} \right)\\
    = \left( {1 + \frac{1}{3} + \frac{1}{5} + ….. + \frac{1}{{99}}} \right) + \left( {\frac{1}{2} + \frac{1}{4} + …. + \frac{1}{{100}}} \right) – 2\left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + …. + \frac{1}{{100}}} \right)\\
    = \left( {1 + \frac{1}{2} + \frac{1}{3} + …. + \frac{1}{{99}} + \frac{1}{{100}}} \right) – \left( {1 + \frac{1}{2} + \frac{1}{3} + ….. + \frac{1}{{50}}} \right)\\
    = 1 + \frac{1}{2} + \frac{1}{3} + …. + \frac{1}{{99}} + \frac{1}{{100}} – 1 – \frac{1}{2} – \frac{1}{3} – ….. – \frac{1}{{50}}\\
    = \frac{1}{{51}} + \frac{1}{{52}} + … + \frac{1}{{100}}.
   \end{array}\)

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