tính nhanh 1+1/1+2+1/1+2+3+1/1+2+3+4+…..1/1+2+3+…+2019

0 bình luận về “tính nhanh 1+1/1+2+1/1+2+3+1/1+2+3+4+…..1/1+2+3+…+2019”

1. Đặt:

   `A = 1 + 1/(1 + 2) + 1/(1+2+3) + 1/(1 + 2 + 3 + 4) .+ … + 1/(1 + 2 + 3 + … + 2019)`

   `A = 1 + (1/(1 + 2) + 1/(1+2+3) + 1/(1 + 2 + 3 + 4) .+ … + 1/(1 + 2 + 3 + … + 2019))`

   Đặt:

   `B = 1/(1 + 2) + 1/(1+2+3) + 1/(1 + 2 + 3 + 4) .+ … + 1/(1 + 2 + 3 + … + 2019)`

   `B = 2/(2 . (1 + 2)) + 2/(2.(1 + 2 + 3)) + 2/(2 . (1 + 2 + 3 + 4)) + … + 2/(2 . (1 + 2 + 3 + … + 2019))`

   `B = 2/(2 . 3) + 2/(2.6) + 2/(2 . 10) + … + 2/(2 . 2039190)`

   `B = 2 . (1/(2.3) + 1/(3.4) + 1/(4.5) + … + 1/(2019 . 2020)`

   `B = 2 . (1/2 – 1/3 + 1/3 – 1/4 + 1/4 – 1/5 + … + 1/2019 – 1/2020)`

   `B = 2 . (1/2 – 1/2020)`

   `B = 2 . 1009/2020`

   `B = 2018/2020`

   `=> A = 1 + 2018/2020`

   `=> A = 2020/2020 + 2018/2020`

   `=> A = 2019/1010`

    

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2. Đáp án + Giải thích các bước giải:

   `1/1+1/(1+2)+1/(1+2+3)+…+1/(1+2+3+…+2019)`

   `=1/((1.(1+1))/2)+1/((2.(2+1))/2)+1/((3.(3+1))/2)+…+1/((2019.(2019+1))/2)`

   `=2/1.2+2/2.3+2/3.4+…+2/2019.2020`

   `=2(1/1.2+1/2.3+1/3.4+…+1/2019.2020)`

   `=2(1-1/2+1/2-1/3+1/3-1/4+…+1/2019-1/2020)`

   `=2(1-1/2020)`

   `=2(2020/2020-1/2020)`

   `=2. 2019/2020`

   `=2019/1010`

    

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