Question

Cho A = \(\left(\dfrac{1}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right).\left(\dfrac{1}{4^2}-1\right)....\left(\dfrac{1}{100^2}-1\right)\)

So sánh A với \(\dfrac{-1}{2}\)

Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\left(\dfrac{1}{4^2}-1\right)..............\left(\dfrac{1}{100^2}-1\right)\)

\(=\left(\dfrac{1}{2}-1\right)\left(\dfrac{1}{2}+1\right)\left(\dfrac{1}{3}-1\right)\left(\dfrac{1}{3}+1\right).............\left(\dfrac{1}{100}-1\right)\left(\dfrac{1}{100}+1\right)\)

\(=\dfrac{-1}{2}.\dfrac{3}{2}.\dfrac{-2}{3}.\dfrac{4}{3}.............\dfrac{-99}{100}.\dfrac{101}{100}\)

\(=\dfrac{-\left(1.2.3....99\right)}{2.3......100}.\dfrac{3.4...101}{2.3....100}\)

\(=\dfrac{-1}{100}.\dfrac{101}{2}\)

\(=\dfrac{-101}{200}< \dfrac{-1}{2}\)

\(\Leftrightarrow A< \dfrac{-1}{2}\)