\(M=\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)...\left(1-\dfrac{1}{900}\right)\)
\(=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{30^2}\right)\)
\(=\left(\dfrac{2^2-1}{2^2}\right)\left(\dfrac{3^2-1}{3^2}\right)\left(\dfrac{4^2-1}{4^2}\right)...\left(\dfrac{30^2-1}{30^2}\right)\)
\(=\left(\dfrac{1.3}{2^2}\right)\left(\dfrac{2.4}{3^2}\right)\left(\dfrac{3.5}{4^2}\right)...\left(\dfrac{29.31}{30^2}\right)\)
\(=\left(\dfrac{1.2.3...29}{2.3.4...30}\right).\left(\dfrac{3.4.5...31}{2.3.4...30}\right)=\dfrac{1}{30}.\dfrac{31}{2}=\dfrac{31}{60}\)
