Cho A = 1/2 x 3/4 x 5/6 x ……. x 99/100. CMR 1/15 < A < 1/10

Giải thích các bước giải:

Ta có :
$\dfrac{n}{n+1}<\dfrac{n+1}{n+2}$

$\to \dfrac 12<\dfrac 23$

      $\dfrac 34<\dfrac 45$

      …..

      $\dfrac{99}{100}<\dfrac{100}{101}$

$\to A=\dfrac 12.\dfrac 34.\dfrac 56…\dfrac{99}{100}<\dfrac 23.\dfrac 45.\dfrac 67…\dfrac{100}{101}$

$\to A^2<\dfrac 12.\dfrac 34.\dfrac 56…\dfrac{99}{100}.\dfrac 23.\dfrac 45.\dfrac 67…\dfrac{100}{101}$

$\to A^2<\dfrac 12.\dfrac 23.\dfrac 34.\dfrac 45.\dfrac 56…\dfrac{99}{100}.\dfrac{100}{101}$

$\to A^2<\dfrac{1}{101}<\dfrac{1}{100}\to A<\dfrac{1}{10}$

Tương tự ta có :

$\dfrac{n}{n+1}<\dfrac{n+1}{n+2}$

$\to \dfrac 12>\dfrac 01$

      $\dfrac 34>\dfrac 23$

      …..

      $\dfrac{99}{100}>\dfrac{98}{99}$

$\to A=\dfrac 12.\dfrac 34.\dfrac 56…\dfrac{99}{100}>\dfrac 12.\dfrac 23.\dfrac 45…\dfrac{98}{99}$

$\to A^2>\dfrac 12.\dfrac 12.\dfrac 23.\dfrac 34.\dfrac 45…\dfrac{98}{99}.\dfrac{99}{100}$

$\to A^2>\dfrac 12.\dfrac{1}{100}=\dfrac{1}{200}>\dfrac{1}{225}\to A>\dfrac{1}{15}$