Ta có :
\(A=\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)\left(1-\dfrac{1}{25}\right)...............\left(1-\dfrac{1}{361}\right)\left(1-\dfrac{1}{400}\right)\)
\(\Rightarrow A=\left(\dfrac{9}{9}-\dfrac{1}{9}\right)\left(\dfrac{16}{16}-\dfrac{1}{16}\right)\left(\dfrac{25}{25}-\dfrac{1}{25}\right).............\left(\dfrac{361}{361}-\dfrac{1}{361}\right)\left(\dfrac{400}{400}-\dfrac{1}{400}\right)\)\(\Rightarrow A=\dfrac{8}{9}.\dfrac{15}{16}.\dfrac{24}{25}................\dfrac{360}{361}.\dfrac{399}{400}\)
\(\Rightarrow A=\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}.\dfrac{4.6}{5^2}...............\dfrac{18.20}{19^2}.\dfrac{19.21}{20^2}\)
\(\Rightarrow A=\dfrac{\left(2.3.4......19\right)\left(4.5.6......21\right)}{\left(3.4.5.....20\right)\left(3.4.5...20\right)}=\dfrac{2.21}{3.20}=\dfrac{7}{10}\)
\(A=\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)\left(1-\dfrac{1}{25}\right)...\left(1-\dfrac{1}{361}\right)\left(1-\dfrac{1}{400}\right)\)
\(A=\dfrac{8}{9}.\dfrac{15}{16}.\dfrac{24}{25}...\dfrac{360}{361}.\dfrac{399}{400}\)
\(A=\dfrac{2.4}{3^2}.\dfrac{3.5}{4^4}.\dfrac{4.6}{5^2}...\dfrac{18.20}{19^2}.\dfrac{19.21}{20^2}\)
\(A=\dfrac{2.3.4...18.19}{3.4.5...20}.\dfrac{4.5.6...21}{3.4.5....20}\)
\(A=\dfrac{1}{10}.7=\dfrac{7}{10}\)
