Tính nhanh tổng sau : M = `4/1.3` + `4/3.5` + `4/5.7` + …..+ `4/2011.2013`

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1. Đáp án:

   $M =\dfrac{4024}{2013}$

   Giải thích các bước giải:

   $\quad M =\dfrac{4}{1.3} +\dfrac{4}{3.5} +\cdots +\dfrac{4}{2011.2013}$

   $\to M = 2\left(\dfrac{2}{1.3} +\dfrac{2}{3.5} +\cdots +\dfrac{2}{2011.2013}\right)$

   $\to M =2\left(\dfrac{3-1}{1.3} +\dfrac{5-3}{3.5} +\cdots +\dfrac{2013-2011}{2011.2013}\right)$

   $\to M = 2\left(1-\dfrac13 +\dfrac13 -\dfrac15 +\cdots +\dfrac{1}{2011} -\dfrac{1}{2013}\right)$

   $\to M = 2\left(1-\dfrac{1}{2013}\right)$

   $\to M =\dfrac{2.2012}{2013}$

   $\to M =\dfrac{4024}{2013}$

   Trả lời

2. M = `4/1.3` + `4/3.5` + `4/5.7` + …….+ `4/2011.2013`

   = 2. ( `2/1.3` + `2/3.5` + …… + `2/2011.2013` )

   = 2 . ( `1/1` – `1/3` + `1/3` – `1/5` + …..+ `1/2011` – `1/2013` )

   = 2. ( 1 – `1/2013` )

   = 2. (  `2013-1/2013` 

   = 2. `2012/2013`

   = `4024/2013`

   Trả lời