Tính A= $\frac{1}{1.3}$ + $\frac{1}{3.5}$ + $\frac{1}{5.7}$ + … + $\frac{1}{97.99}$ + $\frac{1}{99.101}$

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1. Giải thích các bước giải:

   $A = \dfrac{1}{1.3} + \dfrac{1}{3.5} + \dfrac{1}{5.7} + … + \dfrac{1}{97.99} + \dfrac{1}{99.101}$

   $= \dfrac{1}{2}.\left ( \dfrac{2}{1.3} + \dfrac{2}{3.5} + \dfrac{2}{5.7} + … + \dfrac{2}{97.99} + \dfrac{2}{99.101} \right )$

   $= \dfrac{1}{2}.\left ( 1 – \dfrac{1}{3} + \dfrac{1}{3} – \dfrac{1}{5} + \dfrac{1}{5} – \dfrac{1}{7} + … + \dfrac{1}{97} – \dfrac{1}{99} + \dfrac{1}{99} – \dfrac{1}{101} \right )$

   $= \dfrac{1}{2}.\left ( 1 – \dfrac{1}{101} \right )$

   $= \dfrac{1}{2}.\dfrac{100}{101}$

   $= \dfrac{50}{101}$

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2. $A=\dfrac{1}{1.3}$ $+$ $\dfrac{1}{3.5}$ $+$ $\dfrac{1}{5.7}$ $+$ $…$ $+$ $\dfrac{1}{97.99}$ $+$ $\dfrac{1}{99.101}$

     `=“1/2“.“(2/1.3+“2/3.5+“2/5.7+“…+`$\dfrac{2}{97.99}$ $+$ `2/99.101)`

      $=\dfrac{1}{2}.$ `(1-1/3+1/3-1/5+1/5-1/7+…+1/97-1/99+1/99-1/101)`

     `=1/2.(1-1/101)`

     `=1/2` `.` `100/101`

     `=50/101`

   `⇒` `A=50/101`

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