Bài 1: Đặt `A = 1/2^2 + 1/3^3+...+1/100^2`
Ta có: `1/2^2 = 1/2.2 < 1/1.2`
`1/3^2 = 1/3.3 < 1/2.3`
`.........`
`1/100^2 = 1/100.100 < 1/99.100`
`=> 1/2^2 + 1/3^3+...+1/100^2 < 1/1.2 + 1/2.3 +...+ 1/99.100`
`=> A< 1 - 1/2 + 1/2 - 1/3+...+ 1/99-1/100`
`=> A< 1 - 1/100 <1`
Vậy ` 1/2^2 + 1/3^3+...+1/100^2<1`
Bài 2:
a) `( 1/(1.2.3) + 1/(2.3.4) +...+ 1/(8.9.10)) . x = 23/45`
`=> 1/2( 1/1.2 - 1/2.3 + 1/2.3 - 1/3.4+...+1/8.9 - 1/9.10) .x = 23/45`
`=> 1/2 ( 1/1.2- 1/9.10) .x = 23/45`
`=> 1/2 (1/2 - 1/90) .x = 23/45`
`=> 1/2 . 22/45 .x = 23/45`
`=> 11/45 . x = 23/45`
`=> x = 23/45 : 11/45`
`=> x = 23/45 . 45/11`
`=> x=23/11`
Vậy `x= 23/11`
Không tìm được số tự nhiên x