Chứng Minh 1/3-2/3^2+3/3^3-4/3^4+……+99/3^99-100/3^100<3/16

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1. Đáp án:

    

   Giải thích các bước giải:

   →3A=1-2/3+3/3^2-4/3^3+…+99/3^98-100/3^99

   4A=3A+A=(1-2/3+3/3^2-4/3^3+…+99/3^98-100/3^99)+(1/3-2/3^2+…+99/3^99-100/3^100)

   4A=1-1/3+1/3^2-1/3^3+…+1/3^98-1/3^99-100/3^100

   12A=3-1+…+1/3^97-1/3^98-100/3^99

   16A=4A+12A=(1-1/3+1/3^2-1/3^3+…+1/3^98-1/3^99-100/3^100)+(3-1+…+1/3^97-1/3^98-100/3^99)

   16A=3-101/3^99-100/3^100<3

   16A<3

   A<3:16=3/16

   Vậy A<3/16

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2. Đáp án:  $\text { Bạn xem cách làm dưới và THAM KHẢO thôi nhé !! }$

   Giải thích các bước giải:

   Đặt `A = 1/3 – 2/(3^2) + 3/(3^3) – 4/(3^4) + … + 99/(3^99) – 100/(3^100)`

   `⇒ 3A = 1 – 2/3 + 3/(3^2) – 4/(3^3) + … + 99/(3^98) – 100/(3^99)`

   `⇒ 3A + A = 1 – 1/3 + 1/(3^2) – 1/(3^3) + … + 1/(3^99) – 100/(3^100)`

   `⇒ 4A = 1 – 1/3 + 1/(3^2) – 1/(3^3) + … + 1/(3^99) – 100/(3^100)`

   `⇒ 12A = 3 – 1 + 1/3 – 1/(3^2) + … + 1/(3^98) – 100/(3^99)`

   `⇒ 12A + 4A = 3 – 101/(3^99) – 100/(3^100)`

   `⇒ 16A = 3 – 101/(3^99) – 100/(3^100)`

   `⇒ A = (3 – 101/(3^99) – 100/(3^100))/16`

   `⇒ A = 3/16 – (101/(3^99) + 100/(3^100))/16` `< 3/16`

   `⇒ đpcm`

    

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