Cho `C = 1/3 – 2/3^2 + 3/3^3 – 4/3^4 … + 99/3^99 + 100/3^100` CMR : `C < 3/16` (Giúp nhanh nhé !)

Cho `C = 1/3 – 2/3^2 + 3/3^3 – 4/3^4 … + 99/3^99 + 100/3^100`
CMR : `C < 3/16` (Giúp nhanh nhé !)

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  1. Ta có :

    `C=1/3-2/3^2+3/3^3-4/3^4+…+99/3^99-100/3^100`

    `⇔ 3C=1-2/3+3/3^2-4/3^3+…+99/3^98-100/3^99`

    `⇔ 3C+C=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 `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-…-1/3^98`

    `⇔3A+A=3-1/3^99`

    `⇔A=(3-1/3^99)/4`

    `⇔ A=3/4-1/[4.3^99]`

    `⇔4C<3/4-1/[4.3^99]`

    `⇔C<(3/4-1/4.3^99)/4`

    `⇔C<3/16-1/[16.3^99]`

    `⇔C<3/16`

    Xin hay nhất !

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