Question

\(\left(\dfrac{1}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right).\left(\dfrac{1}{4^2}-1\right)...\left(\dfrac{1}{100^2}-1\right)\)

Answer 1

\(\left(\dfrac{1}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right).\left(\dfrac{1}{4^2}-1\right)...\left(\dfrac{1}{100^2}-1\right)\)

\(=\dfrac{-3}{2^2}.\dfrac{-8}{3^2}.\dfrac{-15}{4^2}...\dfrac{-9999}{100^2}\)

\(=-\left(\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}...\dfrac{9999}{100^2}\right)\)

\(=-\left(\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}...\dfrac{99.101}{100.100}\right)\)

\(=-\left(\dfrac{1.2.3.4...99}{2.3.4...100}.\dfrac{3.4.5...101}{2.3.4...100}\right)\)

\(=-\left(\dfrac{1}{100}.\dfrac{101}{2}\right)\)

\(=\dfrac{-101}{200}\)