Question

cho A = (1/4 - 1).(1/9 - 1).(1/16-1).....(1/400 - 1) so sánh A với -1/2

Answer 1

Bài này dễ mà bạn cũng hỏi =(((

\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)....\left(\frac{1}{400}-1\right)\)

\(\Leftrightarrow A=\frac{-3}{4}.\frac{-8}{9}.\frac{-15}{16}....\frac{-399}{400}\)

\(=\frac{1.\left(-3\right)}{2.2}.\frac{2.\left(-4\right)}{3.3}.\frac{3.\left(-5\right)}{4.4}....\frac{19.\left(-21\right)}{20.20}\)

\(=\frac{\left(1.2.3...19\right).\left(\left(-3\right).\left(-4\right).\left(-5\right)...\left(-21\right)\right)}{\left(2.3.4...20\right)\left(2.3.4...20\right)}=\frac{1}{20}.\frac{\left(-21\right)}{2}=\frac{-21}{40}\)

Dễ dàng nhận thấy \(\frac{21}{40}>\frac{1}{2}\Rightarrow\frac{-21}{40}< \frac{-1}{2}\)

Vậy \(A< -\frac{1}{2}\)