Question

Tính

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

Answer 1

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

\(S=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.....\frac{399}{20^2}\)

\(S=\frac{1.3.2.4.3.5......19.21}{2.2.3.3.4.4.....20.20}\)

\(S=\frac{\left(1.2.3.....19\right).\left(3.4.5.....21\right)}{\left(2.3.4....20\right)\left(2.3.4....20\right)}\)

\(S=\frac{21}{20.2}\)

\(S=\frac{21}{40}\)