Chứng minh `1/3-2/3^2+3/3^3-4/3^4+…+99/3^99-100/3^100<3/4`. Giúp em với! Mọi người cho em xin 1 slot thôi ạ!

Question

Chứng minh `1/3-2/3^2+3/3^3-4/3^4+…+99/3^99-100/3^100<3/4`. Giúp em với! Mọi người cho em xin 1 slot thôi ạ!

in progress 0
Vivian 3 tuần 2021-07-11T14:07:48+00:00 2 Answers 2 views 0

Answers ( )

    0
    2021-07-11T14:08:52+00:00

    Đặt:

    `S = 1/3 – 2/3^2 + 3/3^3 – 4/3^4 + … + 99/3^99 – 100/3^100`

    `3S = 3 . ( 1/3 – 2/3^2 + 3/3^3 – 4/3^4 + … + 99/3^99 – 100/3^100)`

    `3S = 1 – 2/3 + 3/3^2 – 4/3^3 + … + 99/3^98 – 100/3^99`

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

    `4S = 1 – 2/3 + 3/3^2 – 4/3^3 + … + 99/3^98 – 100/3^99 + 1/3 – 2/3^2 + 3/3^3 – 4/3^4 + … + 99/3^99 – 100/3^100`

    `4S = 1 – 1/3 + 1/3^2 – 1/3^3 + 1/3^4 – … + 1/3^99 – 100/3^100`

    `=> 4S < 1 – 1/3 + 1/3^2 – 1/3^3 + 1/3^4 – … + 1/3^98 – 1/3^99`

    Lại có:

    Đặt:

    `A = 1 – 1/3 + 1/3^2 – 1/3^3 + 1/3^4 – … + 1/3^98 – 1/3^99`

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

    `3A = 3 – 1 + 1/3 – 1/3^2 + 1/3^3 – … + 1/3^97 – 1/3^98`

    `3A + A = (3 – 1 + 1/3 – 1/3^2 + 1/3^3 – … + 1/3^97 – 1/3^98) – (1 – 1/3 + 1/3^2 – 1/3^3 + 1/3^4 – … + 1/3^98 – 1/3^99)`

    `4A = 3 – 1/3^99`

    `=> 4S < 4A = 3 – 1/3^99`

    `4S < (3 – 1/3^99)/4`

    `4S < 3/4 – 1/(3^99 . 4)`

    `S < 3/4 – 1/(3^99 . 4)`

    `=> S < 3/4`

    Vậy `1/3 – 2/3^2 + 3/3^3 – 4/3^4 + … + 99/3^99 – 100/3^100 < 3/4`

    0
    2021-07-11T14:09:25+00:00

    Đáp án+Giải thích các bước giải:

    `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+1/3^4-…+1/3^98-1/3^99-100/3^100 <1-1/3+1/3^2-1/3^3+1/3^4-…+1/3^98-1/3^99`

    Đặt `B=1-1/3+1/3^2-1/3^3+1/3^4-…+1/3^98-1/3^99`

    `⇔ 3B=3-1+1/3-1/3^2+1/3^3-…-1/3^98`

    `⇔3B+B=3-1/3^99`

    `⇔A=(3-1/3^99)/4`

    `⇔ A=3/4-1/[4.3^99]`

    `⇔4A<3/4-1/[4.3^99]`

    `⇔A<(3/4-1/4.3^99)/4`

    `⇔A<3/4-1/[4.3^99]`

    `⇔A<3/4`

Leave an answer

Browse

35:5x4+1-9:3 = ? ( )