Cho `A = 1/( 1 + 3 ) + 1/( 1 + 3 + 5 ) + 1/( 1 + 3 + 5 + 7 ) + ... + 1/( 1 + 3 + 5 + 7 + ... + 2023)` CMR `A < 3/4`

Ta có `: 1/( 1 + 3 ) + 1/( 1 + 3 + 5 ) + 1/( 1 + 3 + 5 + 7 ) + .... + 1/( 1 + 3 + 5 + 7 + .... + 2023 )`

`= 1/( ( ( 3 + 1 ) . 2 )/2 ) + 1/( ( ( 1 + 5 ) . 3 )/2 ) + 1/( ( ( 7 + 1 ) . 4 )/2 ) + .... + 1/( ( ( 2023 + 1 ) . 1012 )/2 )`

`= 2/( 2 . ( 3 + 1 ) ) + 2/( 3 . ( 1 + 5 ) ) + 2/( 4 . ( 7 + 1 ) ) + .... + 2/( 1012 . ( 2023 + 1 ) )`

`= 2/( 2 . 4 ) + 2/( 3 . 6 ) + 2/( 4 . 8 ) + .... + 2/( 1012 . 2024 )`

`= 1/( 2 . 2 ) + 1/( 3 . 3 ) + 1/( 4 . 4 ) + .... + 1/( 1012 . 1012 )`

`= 1/2^2 + 1/3^2 + 1/4^2 + .... + 1/1012^2`

Ta có `: 1/2^2 = 1/4 , 1/3^2 < 1/( 2 . 3 ) , 1/4^2 < 1/( 3 . 4 ) , .... , 1/1012^2 < 1/( 1011 . 1012 )`

`⇒ 1/2^2 + 1/3^2 + 1/4^2 + .... + 1/1012^2 < 1/4 + 1/( 2 . 3 ) + 1/( 3 . 4 ) + .... + 1/( 1011 . 1012 )`

`⇒ 1/2^2 + 1/3^2 + 1/4^2 + .... + 1/1012^2 < 1/4 + 1/2 - 1/3 + 1/3 - 1/4 + .... + 1/1011 - 1/1012`

`⇒ 1/2^2 + 1/3^2 + 1/4^2 + .... + 1/1012^2 < 1/4 + 1/2 - 1/1012 = 1/4 + 2/4 - 1/1012`

`⇒ 1/2^2 + 1/3^2 + 1/4^2 + .... + 1/1012^2 < 3/4 - 1/1012 < 3/4 (` Điều phải chứng minh `)`