`x + x/( 1 + 2 ) + x/( 1 + 2 + 3 ) + .... + x/( 1 + 2 + 3 + .... + 4041 ) = 4041`

`⇔ x . ( 1 + 1/( 1 + 2 ) + 1/( 1 + 2 + 3 ) + .... + 1/( 1 + 2 + 3 + .... + 4041 )) = 4041`

 `⇔ x . ( 1 + 1/3 + 1/6 + .... + 1/8166861 ) = 4041`

`⇔ x . ( 1 + 2/6 + 2/12 + .... + 2/16333722 ) = 4041`

`⇔ x . ( 1 + 2/( 2 . 3 ) + 2/( 3 . 4 ) + .... + 2/( 4041 . 4042 )) = 4041`

`⇔ x . [ 1 + 2 . ( 1/( 2 . 3 ) + 1/( 3 . 4 ) + .... + 1/( 4041 . 4042 )) ] = 4041`

`⇔ x . [ 1 + 2 . ( 1/2 - 1/3 + 1/3 - 1/4 + .... + 1/4041 - 1/4042 ) ] = 4041`

`⇔ x . [ 1 + 2 . ( 1/2 - 1/4042 ) ] = 4041`

`⇔        x . [ 1 + 2 . 1010/2021 ] = 4041`

`⇔             x . [ 1 + 2020/2021 ] = 4041`

`⇔                        x . 4041/2021 = 4041`

`⇔                                     x        = 4041 : 4041/2021`

`⇔                                     x        = 2021`

Vậy , `x = 2021 .`

Đáp án:

`x + x/(1 + 2) +x/(1 + 2 + 3) + ... + x/(1 + 2 + 3 + ... + 4041) = 4041`

 `⇔ x × [1 + 1/(1 + 2) + 1/(1 + 2 + 3) + ... + 1/(1 + 2 + 3 + ... + 4041)] = 4041 (1)`

`text{Xét :}`

`1/(1 + 2) + 1/(1 + 2 + 3) + ... + 1/(1 + 2 + 3 + ... + 4041)`

`= 1/3 + 1/6 + ... + 1/8166861`

`= (1 × 2)/(3 × 2)  + (1×2)/(6×2) + ... + (1 × 2)/(8166861×2)`

`= 2/6 + 2/12 + ... + 2/16333722`

`= 2/(2 × 3) + 2/(3 × 4) + .... + 2/(4041 × 4042)`

`= 2 × [1/(2 × 3) + 1/(3 × 4) + ... + 1/(4041 × 4042)]`

`= 2 × [1/2 - 1/3 + 1/3 - 1/4 + ... + 1/4041 - 1/4042]`

`= 2 × [1/2 + (- 1/3 + 1/3 - 1/4 + ... + 1/4041) - 1/4042]`

`= 2 × [1/2 - 1/4042]`

`= 2 × 1010/2021`

`= 2020/2021`

`text{Thay vào (1) ta được :}`

`⇔ x × [1 + 2020/2021] = 4041`

`⇔ x × [2021/2021 + 2020/2021] = 4041`

`⇔ x × 4041/2021 = 4041`

`⇔ x = 4041 ÷ 4041/2021`

`⇔ x = 2021`

`text{Vậy x = 2021}`