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 !)

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 !