Toán Chứng tỏ rằng (1-1/3).(1-1/6).(1-1/10).(1-1/15).(1-1/253)<2/5 26/07/2021 By Elliana 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` Trả lời
`\text{Bạn tham khảo cách giải:}` Đặt `A` = `(1 – 1/3)` . `(1 – 1/6)` . `(1 – 1/10)` . `(1 – 1/15)` ……. `(1 – 1/253)` ⇒ `A` = `2/3` . `5/6` . `9/10` . `14/15` ……. `252/253` ⇒ `A` = `4/6` . `10/12` . `18/20` . `28/30` ……. `504/506` ⇒ `A` = `1.4/2.3` . `2.5/3.4` . `3.6/4.5` . `4.7/5.6` ……. `21.24/22.23` ⇒ `A` = `1.4.2.5.3.6.4.7……..21.24/2.3.3.4.4.5.5.6…….22.23` ⇒ `A` = `((1.2.3…….21).(4.5.6……..24))/((2.3..4…….2).(3.4.5…….23))` ⇒ `A` = `1.24/22.3` ⇒ `A` = `4/11` `\text{Ta có:}` `4/11` = `20/55` `2/5` = `22/55` ⇒ `20/55` < `22/55` ⇒ `4/11` < `2/5` ⇒ `A` < `2/5` (đpcm) `\text{Vậy A <}` `2/5`. #NOCOPY Trả lời
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`
`\text{Bạn tham khảo cách giải:}`
Đặt `A` = `(1 – 1/3)` . `(1 – 1/6)` . `(1 – 1/10)` . `(1 – 1/15)` ……. `(1 – 1/253)`
⇒ `A` = `2/3` . `5/6` . `9/10` . `14/15` ……. `252/253`
⇒ `A` = `4/6` . `10/12` . `18/20` . `28/30` ……. `504/506`
⇒ `A` = `1.4/2.3` . `2.5/3.4` . `3.6/4.5` . `4.7/5.6` ……. `21.24/22.23`
⇒ `A` = `1.4.2.5.3.6.4.7……..21.24/2.3.3.4.4.5.5.6…….22.23`
⇒ `A` = `((1.2.3…….21).(4.5.6……..24))/((2.3..4…….2).(3.4.5…….23))`
⇒ `A` = `1.24/22.3`
⇒ `A` = `4/11`
`\text{Ta có:}` `4/11` = `20/55`
`2/5` = `22/55`
⇒ `20/55` < `22/55`
⇒ `4/11` < `2/5`
⇒ `A` < `2/5` (đpcm)
`\text{Vậy A <}` `2/5`.
#NOCOPY