Chứng tỏ rằng (1-1/3).(1-1/6).(1-1/10).(1-1/15).(1-1/253)<2/5 26/07/2021 Bởi Iris Chứng tỏ rằng (1-1/3).(1-1/6).(1-1/10).(1-1/15).(1-1/253)<2/5
Đáp án: `(1 – 1/3) (1 – 1/6) (1 – 1/10) (1 – 1/15) … (1 – 1/253)` `= 2/3 × 5/6 × 9/10 × 14/15 × … × 252/253` `= (2 × 2)/(3 × 2) × (5 × 2)/(6 × 2) × (9 × 2)/(10 × 2) × (14 × 2)/(15 × 2) × … × (252 × 2)/(253 × 2)` `= 4/6 × 10/12 × 18/20 × 28/30 × … × 504/506` `= (4 × 10 × 18 × 28 × … × 504)/(6 × 12 × 20 × 30 × … × 506)` `= (1 × 2×3×…×21)/(2×3×4…×22) × (4 ×5×6×…×24)/(3×4×5×…×23)` `= 1/22 × 24/3` `= 24/66` `= 4/11` `text{Ta thấy :}` `4/11 < 2/5` `-> (1 – 1/3) (1 – 1/6) (1 – 1/10) (1 – 1/15) … (1 – 1/253) < 2/5` Bình luận
Đáp án: Đặt `A=(1-1/3).(1-1/6).(1-1/10).(1-1/15)….(1-1/253)` `= 2/3 . 5/6 . 9/10 . 14/15 …. 252/253` `= 4/6 . 10/12 . 18/20 . 28/30 … 504/506` `= (4.10.18.28…504)/(6.12.20.30…506)` `= (1.4.2.5.3.6.4.7…21.24)/(2.3.3.4.4.5.5.6…22.23)` `= (1.2.3…21)/(2.3.4…22) . (4.5.6….24)/(3.4.5….23)` `= 1/22 . 24/3 = 4/11 < 22/55 = 2/5` Vậy `A<2/5` Bình luận
Đáp án:
`(1 – 1/3) (1 – 1/6) (1 – 1/10) (1 – 1/15) … (1 – 1/253)`
`= 2/3 × 5/6 × 9/10 × 14/15 × … × 252/253`
`= (2 × 2)/(3 × 2) × (5 × 2)/(6 × 2) × (9 × 2)/(10 × 2) × (14 × 2)/(15 × 2) × … × (252 × 2)/(253 × 2)`
`= 4/6 × 10/12 × 18/20 × 28/30 × … × 504/506`
`= (4 × 10 × 18 × 28 × … × 504)/(6 × 12 × 20 × 30 × … × 506)`
`= (1 × 2×3×…×21)/(2×3×4…×22) × (4 ×5×6×…×24)/(3×4×5×…×23)`
`= 1/22 × 24/3`
`= 24/66`
`= 4/11`
`text{Ta thấy :}` `4/11 < 2/5`
`-> (1 – 1/3) (1 – 1/6) (1 – 1/10) (1 – 1/15) … (1 – 1/253) < 2/5`
Đáp án:
Đặt `A=(1-1/3).(1-1/6).(1-1/10).(1-1/15)….(1-1/253)`
`= 2/3 . 5/6 . 9/10 . 14/15 …. 252/253`
`= 4/6 . 10/12 . 18/20 . 28/30 … 504/506`
`= (4.10.18.28…504)/(6.12.20.30…506)`
`= (1.4.2.5.3.6.4.7…21.24)/(2.3.3.4.4.5.5.6…22.23)`
`= (1.2.3…21)/(2.3.4…22) . (4.5.6….24)/(3.4.5….23)`
`= 1/22 . 24/3 = 4/11 < 22/55 = 2/5`
Vậy `A<2/5`