Đáp án:
`A<3/4`
Giải thích các bước giải:
`A=1/3+2/3^2+3/3^3+4/3^4+...+99/3^99+100/3^100`
`=>3A=3(1/3+2/3^2+3/3^3+4/3^4+...+99/3^99+100/3^100)`
`=>3A=1/(3÷3)+2/(3^2÷3)+3/(3^3÷3)+4/(3^4÷3)+...+99/(3^99÷3)+100/(3^100÷3)`
`=>3A=1+2/3+3/3^2+4/3^3+...+99/3^98+100/3^99`
`=>3A-A=(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)`
`=>2A=1+1/3+1/3^2+1/3^3+...+1/3^99+100/3^100`
`=>6A=3(1+1/3+1/3^2+1/3^3+...+1/3^99+100/3^100)`
`=>6A=3+1/(3÷3)+1/(3^2÷3)+1/(3^3÷3)+...+1/(3^99÷3)+100/(3^100÷3)`
`=>6A=3+1+1/3+1/3^2+...+1/3^98+100/(3^99)`
`=>6A-2A=(3+1+1/3+1/3^2+...+1/3^98+100/(3^99))-(1+1/3+1/3^2+1/3^3+...+1/3^99+100/3^100)`
`=>4A=3-100/3^99+100/3^100`
`=>`$A=\dfrac{3-\dfrac{100}{3^{100}}+\dfrac{101}{3^{101}}}{4}$
`=>`$A=\dfrac{3}{4}-\dfrac{\dfrac{100}{3^{100}}+\dfrac{101}{3^{101}}}{4}<\dfrac34$
Vậy `A<3/4`.
