Đáp án:
`8, 1011/2023`
`9, 11/100`
`10, 41/250`
Giải thích các bước giải:
`8,`
`1/(1×3) + 1/(3×5) + ... + 1/(2021 × 2023)`
`= 1/2 × [1 - 1/3 + 1/3 - 1/5 + ... + 1/2021 - 1/2023]`
`= 1/2 × [1 + (-1/3 + 1/3) + ... + (-1/2021 + 1/2021) - 1/2023]`
`= 1/2 × [1 - 1/2023]`
`= 1/2 × [2023/2023 - 1/2023]`
`= 1/2 × 2022/2023`
`= 1011/2023`
`9.`
`1/(1×10) + 1/(10 × 19) + ... + 1/(91 ×100)`
`= 1/9 × [1 - 1/10 + 1/10 - 1/19 + ... + 1/91 - 1/100]`
`= 1/9× [1 + (-1/10 + 1/10) + ... + (-1/91 + 1/91) - 1/100]`
`= 1/9 × [1 - 1/100]`
`= 1/9 × [100/100 - 1/100]`
`= 1/9 × 99/100`
`= 11/100`
`10.`
`1/(2×5) + 1/(5×8) + ... + 1/(122 × 125)`
`= 1/3 × [1/2 - 1/5 + 1/5 - 1/8 + ... + 1/122 - 1/125]`
`= 1/3 × [1/2 + (-1/5 + 1/5) + ... +(-1/122+1/122) - 1/125]`
`= 1/3 × [1/2 - 1/125]`
`= 1/3 × [125/250 - 2/250]`
`= 1/3 × 123/250`
`= 41/250`