Cho C= 1/102+1/103+......+1/200 . Chứng minh rằng C>7/12

Ta có :

`C = 1/101 + 1/102 + 1/103 + .... + 1/200`

` = (1/101 + 1/102 + 1/103 + .... + 1/149 + 1/150) + (1/151 + 1/152 + 1/153 + .... +1/199+ 1/200)`

Ta thấy :

`1/101 > 1/150`

`1/102 > 1/150`

`1/103 > 1/150`

`....`

`1/149 > 1/150`

`1/150 =1/150`

`=> 1/101 + 1/102 + 1/103 + ... + 1/150 > 1/150 + 1/150 + 1/150 + ... + 1/150` (có `50` số hạng)

`=> 1/101 + 1/102 + 1/103 + ... + 1/150 >  1/150 . 50`

`=> 1/101 + 1/102 + 1/103 + ... + 1/150 > 1/3 (1)`

Ta thấy :

`1/151 > 1/200`

`1/152 > 1/200`

`1/153 > 1/200`

`....`

`1/199 > 1/200`

`1/200 = 1/200`

`=> 1/151 + 1/152 + 1/153 + .... +1/199+ 1/200 > 1/200 + 1/200 + 1/200 + .... + 1/200` (có `50` số hạng)

`=> 1/151 + 1/152 + 1/153 + .... +1/199+ 1/200 > 1/200 . 50`

`=> 1/151 + 1/152 + 1/153 + .... +1/199+ 1/200 >  1/4 (2)`

Từ `(1)` và `(2)` suy ra :

`(1/101 + 1/102 + 1/103 + .... + 1/149 + 1/150) + (1/151 + 1/152 + 1/153 + .... +1/199+ 1/200) >  1/3 + 1/4`

`=> C > 7/12`