Chứng tỏ rằng (1-1/3).(1-1/6).(1-1/10).(1-1/15).(1-1/253)<2/5

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1. 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`

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2. `\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 

    

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