lớp 8a3 nguyễn khuyến đúng ko

\(B=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right).....\left(1-\frac{1}{100}\right)\)

\(=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...............\frac{99}{100}\)

\(=\frac{3.8.15......99}{4.9.16....100}=\frac{\left(1.3\right).\left(2.4\right).\left(3.5\right).......\left(9.11\right)}{\left(2.2\right).\left(3.3\right).\left(4.4\right)......\left(10.10\right)}\)

\(=\frac{\left(1.2.3.....9\right).\left(3.4.5......11\right)}{\left(2.3.4.....10\right).\left(2.3.4.......10\right)}=\frac{1}{10}.\frac{11}{2}=\frac{11}{20}< \frac{11}{21}\)

Vậy B<11/21

11/20 >11/21

\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right).......\left(1-\frac{1}{19}\right)\left(1-\frac{1}{20}\right)\) 

 \(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}......\frac{18}{19}.\frac{19}{20}\)

\(A=\frac{1}{20}\)

\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)........\left(1-\frac{1}{19}\right)\left(1-\frac{1}{20}\right)\)

\(\Leftrightarrow A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...........\frac{18}{19}.\frac{19}{20}\)

\(\Leftrightarrow A=\frac{1}{20}>\frac{1}{21}\)

\(\Leftrightarrow A>\frac{1}{21}\)

\(B=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)................\left(1-\frac{1}{100}\right)\)

\(\Leftrightarrow B=\frac{3}{4}.\frac{8}{9}..................\frac{99}{100}\)

\(B=\frac{1.3}{2^2}.\frac{2.4}{3^2}................\frac{9.11}{50^2}\)

\(B=\frac{11}{50}< \frac{11}{21}\)