\(\dfrac{9}{9.19}+\dfrac{9}{19.29}+...+\dfrac{1}{89.99}\)
\(=9.\left(\dfrac{1}{9.19}+\dfrac{1}{19.29}+...+\dfrac{1}{79.89}\right)+\dfrac{1}{89.99}\)
\(=\dfrac{9}{10}.\left(\dfrac{10}{9.19}+\dfrac{10}{19.29}+...+\dfrac{10}{79.89}\right)+\dfrac{1}{89.99}\)
\(=\dfrac{9}{10}.\left(\dfrac{1}{9}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{29}+...+\dfrac{1}{79}-\dfrac{1}{89}\right)+\left(\dfrac{1}{89}-\dfrac{1}{99}\right)\)
\(=\dfrac{9}{10}.\left(\dfrac{1}{9}-\dfrac{1}{89}\right)+\dfrac{10}{8811}\)
\(=\dfrac{9}{10}.\dfrac{80}{801}+\dfrac{10}{8811}\)
\(=\dfrac{8}{89}+\dfrac{10}{8811}\)
\(=\dfrac{802}{8811}\)
Đề là `9/[89.99]` thay vì là `1/[89.99]` chứ nhỉ? Vì nếu bạn để đề vậy thì không thấy quy luật là như thế nào.
*Sửa: `9/[9.19]+9/[19.29]+....+9/[89.99]`
`=9/10(10/[9.19]+10/[19.29]+....+10/[89.99])`
`=9/10(1/9-1/19+1/19-1/29+.....+1/89-1/99)`
`=9/10(1/9-1/99)`
`=9/10 . 10/99`
`=1/11`