`M=20/(1×5)+25/(5×10)+30/(10×16)+35/(16×23)+40/(23×31)`

`M=5×[4/(1×5)+5/(5×10)+6/(10×16)+7/(16×23)+8/(23×31)]`

`M=5×(1-1/5+1/5-1/10+1/0-1/16+1/16-1/23+1/23-1/31]`

`M=5×[1-1/31]`

`M=5×30/31`

`M=150/31`

`b`)

 `A=1/2^2+1/3^2+1/4^2+...+1/2002^2+1/2003^2`

Ta có:

`1/2^2<1/(1×2)`

`1/3^2<1/(2×3)`

`1/4^2<1/(3×4)`

....

`1/2002^2<1/(2001×2002)`

`1/2003^2<1/(2002×2003)`

`⇒ A<1/(1×2)+1/(2×3)+1/(3×4).+1/(2001×2002)+1/(2002×2003)`

`⇒ A<1-1/2+1/2-1/3+1/3-1/4+1/4-....+1/2001-1/2002+1/2002-1/2023`

`⇒ A<1-1/2023`

Mà `1-1^2023<1`

`⇒A<1`