S= 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/16 + 1/24 +.......+ 1/256 + 1/384

Answer

`S = 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/16 + 1/24 + ..... + 1/256 + 1/384`

`S = (1/2 + 1/4 + 1/8 + 1/16 + ..... + 1/256) + (1/3 + 1/6 + 1/12 + 1/24 + .... + 1/384)`

Đặt `A = 1/2 + 1/4 + 1/8 + 1/16 + ..... + 1/256`

$\bullet$ Ta có: `A = 1/2 + 1/4 + 1/8 + 1/16 + ..... + 1/256`

`=> 2A = 1 + 1/2 + 1/4 + 1/8 + .... + 1/128`

`=> 2A - A = (1 + 1/2 + 1/4 + 1/8 + .... + 1/128) - (1/2 + 1/4 + 1/8 + 1/16 + ..... + 1/256)`

`=> A = 1 - 1/256`

`=> A = 255/256`

Đặt `B = 1/3 + 1/6 + 1/12 + 1/24 + .... + 1/384`

$\bullet$ Ta có: `B = 1/3 + 1/6 + 1/12 + 1/24 + .... + 1/384`

`B = 1/3 . (1 + 1/2 + 1/4 + 1/8 + ..... + 1/128)`

`=> B/3 = 1 + 1/2 + 1/4 + 1/8 + ..... + 1/128`

`=> (2B)/3 = 2 + 1 + 1/2 + 1/4 + ..... + 1/64`

`=> (2B)/3 - B/3 = (2 + 1 + 1/2 + 1/4 + ..... + 1/64) - (1 + 1/2 + 1/4 + 1/8 + ..... + 1/128)`

`=> B/3 = 2 - 1/128`

`=> B = 1/3 . 255/128`

`=> B = 85/128`

`=> S = A + B`

`=> S = 255/256 + 85/128`

`=> S = 255/256 + 170/256`

`=> S = 425/256`

Vậy `S = 425/256`