\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)....................\left(\dfrac{1}{10^2}-1\right)\)
\(=\left(\dfrac{1}{4}-\dfrac{4}{4}\right)\left(\dfrac{1}{9}-\dfrac{9}{9}\right)...........\left(\dfrac{1}{100}-\dfrac{100}{100}\right)\)
\(=\dfrac{-3}{4}.\dfrac{-8}{9}..............\dfrac{-99}{100}\)
\(=\dfrac{\left(-1\right).3}{2.2}.\dfrac{\left(-2\right).4}{3.3}..................\dfrac{\left(-9\right).11}{10.10}\)
\(=\dfrac{\left(-1\right)\left(-2\right)..........\left(-9\right)}{2.3.....10}.\dfrac{3.4....11}{2.3....10}\)
\(=\dfrac{-1}{10}.\dfrac{11}{2}\)
\(=\dfrac{-11}{20}< \dfrac{-10}{20}=\dfrac{-1}{2}\)
\(\Leftrightarrow A< \dfrac{-1}{2}\)
\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)...\left(\dfrac{1}{10^2}-1\right)\)
\(A=\left(\dfrac{1}{4}-1\right)\left(\dfrac{1}{9}-1\right).....\left(\dfrac{1}{100}-1\right)\)
\(A=\left(\dfrac{1}{4}-\dfrac{4}{4}\right)\left(\dfrac{1}{9}-\dfrac{9}{9}\right)....\left(\dfrac{1}{100}-\dfrac{100}{100}\right)\)
\(A=\dfrac{-3}{4}.\dfrac{-8}{9}.....\dfrac{-99}{100}\)
\(A=\dfrac{\left(-1\right).3}{4}.\dfrac{\left(-1\right).8}{9}......\dfrac{\left(-1\right).99}{100}\)
\(A=\dfrac{\left(-1\right).1.3}{2.2}.\dfrac{-1.2.4}{3.3}....\dfrac{-1.9.11}{10.10}\)
\(A=\dfrac{-1.3}{2.2}.\dfrac{-2.4}{3.3}....\dfrac{-9.11}{10.10}\)
\(A=\dfrac{\left(-1\right)\left(-2\right)....\left(-9\right)}{2.3.....10}.\dfrac{3.4....11}{2.3.....10}\)
\(A=\dfrac{-1}{10}.\dfrac{11}{2}=-\dfrac{11}{20}\)
