Đáp án:
$\\$
Bài `2`
`A = (-1)/20 + (-1)/30 + ... + (-1)/72 + (-1)/90`
`↔ A = - 1/20 - 1/30 - ... - 1/72 - 1/90`
`↔ A = - [1/20 + 1/30 + ... + 1/72 + 1/90]`
`↔ A = -[1/(4×5) + 1/(5×6) + ... + 1/(8 × 9) + 1/(9 × 10)]`
`↔ A = - [1/4 - 1/5 + 1/5 - 1/6 + ... + 1/8 -1/9 + 1/9 - 1/10]`
`↔ A = -[1/4 + ( - 1/5 + 1/5 - 1/6 + ... + 1/8 -1/9 + 1/9) - 1/10]`
`↔ A = - [1/4 - 1/10]`
`↔ A = (-3)/20`
Vậy `A = (-3)/20`
$\\$
Bài `3`
$\bullet$ `A = 2/(60 × 63) + 2/(63 × 66) + ...+ 2/(117 × 120) + 2/2011`
`↔ A = 2/3 × [1/60 - 1/63 + 1/63 - 1/66 + ... + 1/117 - 1/120] + 2/2011`
`↔ A = 2/3 × [1/60 + (- 1/63 + 1/63 - 1/66 + ... + 1/117) - 1/120] + 2/2011`
`↔ A = 2/3 × [1/60 - 1/120] + 2/2011`
`↔ A = 2/3 × 1/120 + 2/2011`
`↔ A = 1/180 + 2/2011`
$\bullet$ `B = 5/(40 × 44) + 5/(44 × 48) + .. + 5/(76 × 80) + 5/2011`
`↔ B = 5/4 × [1/40 - 1/44 + 1/44 - 1/48 + ... + 1/76 - 1/80] + 5/2011`
`↔ B = 5/4 × [1/40 + (- 1/44 + 1/44 - 1/48 + ... + 1/76) - 1/80] + 5/2011`
`↔ B = 5/4 × [1/40 - 1/80] + 5/2011`
`↔ B = 5/4 × 1/80 + 5/2011`
`↔ B = 1/64 + 5/2011`
Có : \(\left\{ \begin{array}{l}A = \dfrac{1}{180} + \dfrac{2}{2011}\\B=\dfrac{1}{64} + \dfrac{5}{2011}\end{array} \right.\)
Ta thấy : `2/2011 < 5/2011` (Do `2 < 5`) `(1)`
Ta có : `180 > 64`
`-> 1/180 < 1/64` `(2)`
Từ `(1), (2)`
`-> 1/180 + 2/2011 < 1/64 + 5/2011`
`-> A < B`
Vậy `A < B`
