Question

Cho A= (1/2^2 - 1) (1/3^2 - 1) (1/4^2 - 1) ...(1/100^2 - 1)

So sánh A với -1/2

 

Answer 1

Ta có :

\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)

\(\Leftrightarrow A=-\frac{3}{4}.-\frac{8}{9}....-\frac{9999}{10000}\)

\(\Leftrightarrow A=\frac{-3.8....9999}{4.9.10000}=\frac{-3.2.4.....99.101}{2.2.3.3....100.100}=\frac{-101}{100}\)

Mà \(-\frac{1}{2}=-\frac{50}{100}>-\frac{101}{100}\) 

Vậy A < \(-\frac{1}{2}\)