Question

Tìm tích:

\(\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)....\left(1-\frac{1}{1999^2}\right)\left(1-\frac{1}{2000^2}\right)\)

Answer 1

\(\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)..\left(1-\frac{1}{2000^2}\right)\)

 

\(=\frac{1.3}{2^2}\cdot\frac{2.4}{3^2}\cdot\frac{3.5}{4^2}\cdot\cdot\cdot\cdot\frac{1998.2000}{1999^2}\cdot\frac{1999.2001}{2000^2}\)

\(=\frac{1}{2}\cdot\frac{2001}{2000}=\frac{2001}{4000}\)

Answer 2

\(\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{1999^2}\right)\left(1-\frac{1}{2000^2}\right)\)

=\(\left(\frac{4}{4}-\frac{1}{4}\right)\left(\frac{9}{9}-\frac{1}{9}\right)...\left(\frac{3996001}{3996001}-\frac{1}{3996001}\right)\left(\frac{4000000}{4000000}-\frac{1}{4000000}\right)\)

=\(\frac{3}{4}.\frac{8}{9}....\frac{3996000}{3996001}.\frac{3999999}{4000000}\)

=\(\frac{1.3}{2.2}.\frac{2.4}{3.3}...\frac{1998.2000}{1999.1999}.\frac{1999.2001}{2000.2000}\)

=\(\frac{1.3.2.4.3.6.....1998.2000.1999.2001}{2.2.3.3.4.4....1999.1999.2000.2000}=\frac{1.2001}{2.2000}=\frac{2001}{4000}\)