Đáp án `+` Giải thích các bước giải`:`

Ta có`:`

`P=(2^2)/1.3+(4^2)/3.5+(6^2)/5.7+(8^2)/7.9+...+(2024^2)/2023.2025`

`P=2.2/1.3+4.4/3.5+6.6/5.7+8.8/7.9+...+2024.2024/2023.2025`

`P=(1.3+1)/1.3+(3.5+1)/3.5+(5.7+1)/5.7+(7.9+1)/7.9+...+(2023.2025+1)/2023.2025`

`P=1.3/1.3+1/1.3+3.5/3.5+1/3.5+5.7/5.7+1/5.7+7.9/7.9+1/7.9+...+2023.2025/2023.2025+1/2023.2025`

`P=1+1/1.3+1+1/3.5+1+1/5.7+1+1/7.9+...+1+1/2023.2025`

`P=1012+(1/1.3+1/3.5+1/5.7+1/7.9+...+1/2023.2025)`

Đặt `A=1/1.3+1/3.5+1/5.7+1/7.9+...+1/2023.2025`

`2A=2/1.3+2/3.5+2/5.7+2/7.9+...+2/2023.2025`

`2A=1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+...+1/2023-1/2025`

`2A=1-1/2025`

`A=1/2-1/4050`

Thay `A` vào `P,` ta có`:`

`P=1012+1/2-1/4050`

`P=1012+0,5-1/4050`

`P=1012,5-1/4050`

Do đó `P<1012,5`

Vậy `P<1012,5` (đpcm)