`\text{Đề này sai nhé, phải là:}`

`1` `-` `1/3` `+` `1/9` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `3/3` `-` `1/3` `+` `1/9` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `2/3` `+` `1/9` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `6/9` `+` `1/9` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `7/9` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `21/27` `-` `1/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `20/27` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `60/81` `+` `1/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `61/81` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `183/243` `-` `1/243` `+` `1/729` `-` `1/2187`

`=` `182/243` `+` `1/729` `-` `1/2187`

`=` `1638/2187` `+` `3/2187` `-` `1/2187`

`=` `\frac{1638+3-1}{2187}`

`=` `1640/2187`

@stipud

 

Đáp án:  Mình sửa lại thành `1/2187` ở cuối nhé

Giải thích các bước giải:

`A = 1/2 - 1/3 - 1/9 - .... - 1/729 - 1/2187`

`A = 1/2 - (1/3 + 1/9 + .... - 1/729 + 1/2187)`

`A = 1/2 - ( 1/3 + 1/(3^2) + ... + 1/(3^6) + 1/(3^7) )`

Đặt `B = 1/3 + 1/(3^2) + ... + 1/(3^6) + 1/(3^7)`

`⇒ 3B = 1 + 1/3 + ... + 1/(3^5) + 1/(3^6)`

`⇒ 3B - B = 1 - 1/(3^7)`

`⇒ 2B = 1 - 1/(3^7)`

`⇒ B = (1 - 1/(3^7))/2`

`⇒ B = 1/2 - 1/(3^7 . 2)`

Như vậy, ta có: `A = 1/2 - 1/2 + 1/(3^7 . 2)`

     `⇒ A = 1/(3^7 . 2)`

     `⇒ A = 1/4374`