\(\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)...........\left(\dfrac{1}{100^2}-1\right)\)
\(=\left(\dfrac{1}{4}-1\right)\left(\dfrac{1}{9}-1\right).......\left(\dfrac{1}{10000}-1\right)\)
\(=\dfrac{-3}{4}.\dfrac{-8}{9}...........\dfrac{-9999}{10000}\)
\(=\dfrac{\left(-1\right).3}{2^2}.\dfrac{\left(-2\right).4}{3^2}.................\dfrac{\left(-99\right).101}{100^2}\)
\(=\dfrac{\left(-1\right)\left(-2\right)........\left(-99\right)}{2.3.4.....100}.\dfrac{3.4...101}{2.3.4...100}\)
\(=\dfrac{-1}{100}.\dfrac{101}{2}\)
\(=\dfrac{-101}{200}\)
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