cho S =5/2^2+5/3^2+5/4^2+...+5/100^2 CMR 2<S<5

Ta có

$S = \dfrac{5}{2^2}  + \dfrac{5}{3^2} + \cdots + \dfrac{5}{100^2}$

$> \dfrac{5}{2.3} + \dfrac{5}{3.4} + \cdots + \dfrac{ 5}{100.101}$

$ = \dfrac{5}{2} - \dfrac{5}{3} + \dfrac{5}{3} - \dfrac{5}{4} + \cdots + \dfrac{5}{100} - \dfrac{5}{101}$

$= \dfrac{5}{2} - \dfrac{5}{101}$

$> \dfrac{5}{2} - \dfrac{1}{2} = 2$

Vậy $S > 2$.

Mặt khác, ta lại có

$S = \dfrac{5}{2^2}  + \dfrac{5}{3^2} + \cdots + \dfrac{5}{100^2}$

$< \dfrac{5}{1.2} + \dfrac{5}{2.3} + \cdots + \dfrac{5}{99.100}$

$= \dfrac{5}{1} - \dfrac{5}{2} + \dfrac{5}{2} - \dfrac{5}{3} + \cdots + \dfrac{5}{99} - \dfrac{5}{100}$

$= 5 - \dfrac{5}{100} < 5$

Vậy $S < 5$

Suy ra $2 < S < 5$.