tính A=(1-1/22) . (1-1/32) . ……….. . (1-1/20202)

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tính A=(1-1/22) . (1-1/32) . ……….. . (1-1/20202)

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Anna 3 tuần 2021-11-25T19:46:29+00:00 1 Answers 2 views 0

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    2021-11-25T19:48:25+00:00

    $A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{2020^2})$

    Các số trên có dạng 

    $1-\dfrac{1}{a^2} = \dfrac{a^2-1}{a^2} = \dfrac{(a-1).(a+1)}{a^2}$

    Do đó $A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{2020^2})$

    $ = \dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}….\dfrac{2019.2021}{2020.2020}$

    $ = \dfrac{(1.2.3…2019).(3.4…2021)}{2^2.3^2.4^2….2020^2}$

    $ = \dfrac{2021}{2020.2}$

    $  =\dfrac{2021}{4040}$

     

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