So sánh A và B A= (1- 1/2)(1- 1/3)(1- 1/4) … (1- 1/20) B= (1- 1/4)(1- 1/9)(1- 1/16)…(1- 1/80)(1- 1/100)

0 bình luận về “So sánh A và B A= (1- 1/2)(1- 1/3)(1- 1/4) … (1- 1/20) B= (1- 1/4)(1- 1/9)(1- 1/16)…(1- 1/80)(1- 1/100)”

1. Đáp án:

    `A<B`

   Giải thích các bước giải:

    Xét `A`

   `⇒A=(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})….(1-\frac{1}{19})(1-\frac{1}{20})`

   `⇒A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}…..\frac{18}{19}.\frac{19}{20}`

   `⇒A=\frac{1.(2.3……19)}{(2.3.4…..19).20}`

   `⇒A=\frac{1}{20}<\frac{1}{2}(1)`

   Xét `B`

   `⇒B=(1-\frac{1}{4})(1-\frac{1}{9})(1-\frac{1}{16})…(1-\frac{1}{100^2})`

   `⇒B=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}….\frac{9999}{100^2}`

   `⇒B=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}….\frac{99.101}{100^2}`

   `⇒B=\frac{1.2.3…98.99}{2.3.4…99.100}.\frac{3.4.5…100.101}{2.3.4…100}`

   `⇒B=\frac{1}{100}.\frac{101}{2}=\frac{101}{200}>\frac{1}{2}(2)`

   Từ `(1)(2)⇒A<B`

   Bình luận

2. Ta có:

   `A = (1 – 1/2) . (1 – 1/3) . (1 – 1/4) . …   . (1 – 1/20)`

   `A = 1/2 . 2/3 . 3/4 . …  . 19/20`

   `A = (1 . 2 . 3 . …   . 19)/(2 . 3 . 4 . …  . 20)`

   `A = 1/20`

   $\\$

   `B = (1 – 1/4) . (1 – 1/9) . (1 – 1/16) . … . (1 – 1/80) . (1 – 1/100)`

   `B = 3/4 . 8/9 . 15/16 . …   . 79/80 . 99/100`

   `B = (3 . 8 . 15 . …  . 79 . 99)/(4 . 9 . 16 . …  . 80 . 100)`

   `B = (1 . 3 . 2 . 4 . 3 . 5 . … . 9 . 11)/(2 . 2 . 3 . 3 . 4 . 3 . … . 10 . 10)`

   `B = ((1 . 2 . 3 . 4 . 5 . … . 9 . 10) . (3 . 4 . 5 . …  . 11))/(2 . 3 . 4 . … . 10) . (2 . 3 . 4 . …  . 10)`

   `B = 11/(10 . 2)`

   `B = 11/20`

   Mà `1/20 < 11/20`

   `=> A < B`

   Vậy `A < B`

   Bình luận