so sánh giá trị biểu thức A=1/2.9+1/3.12+1/4.15+1/5.18+…+1/2020.6063 với 1/6

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1. Đáp án: $A < \dfrac{1}{6}$

   Giải thích các bước giải:

   $\begin{array}{l}
   A = \dfrac{1}{{2.9}} + \dfrac{1}{{3.12}} + \dfrac{1}{{4.15}} + \dfrac{1}{{5.18}} + … + \dfrac{1}{{2020.6063}}\\
    = \dfrac{1}{3}.\left( {\dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + \dfrac{1}{{4.5}} + \dfrac{1}{{5.6}} + … + \dfrac{1}{{2020.2021}}} \right)\\
    = \dfrac{1}{3}.\left( {\dfrac{1}{2} – \dfrac{1}{3} + \dfrac{1}{3} – \dfrac{1}{4} + \dfrac{1}{4} – \dfrac{1}{5} + … + \dfrac{1}{{2020}} – \dfrac{1}{{2021}}} \right)\\
    = \dfrac{1}{3}.\left( {\dfrac{1}{2} – \dfrac{1}{{2021}}} \right) < \dfrac{1}{3}.\dfrac{1}{2} = \dfrac{1}{6}
   \end{array}$

    Vậy $A < \dfrac{1}{6}$

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2. Đáp án:

   `A<1/6`

   Giải thích các bước giải:

   `A=1/2.9+1/3.12+1/4.15+1/5.18+…+1/2020.6063`

   `3A=(1.3)/2.9+(1.3)/3.12+(1.3)/(4.15)+(1.3)/5.18+…+(1.3)/2020.6063`

   `3A=1/2.3+1/3.4+1/4.5+1/5.6+…+1/2020.2021`

   `3A=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+…+1/2020-1/2021`

   `3A=1/2-1/2021`

   `3A=2019/4042`

   `A=2019/4042:3`

   `A=673/4042`

   `A=673/4042<673/4038=1/6`

   Vậy `A<1/6`.

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