Question

Answer 1

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

\(D=\left(\dfrac{1-4}{2^2}\right)\left(\dfrac{1-9}{3^2}\right)\left(\dfrac{1-16}{4^2}\right)....\left(\dfrac{1-1000}{100^2}\right)\)

\(D=\dfrac{-3}{2^2}.\dfrac{-8}{3^2}.\dfrac{-15}{4^2}.....\dfrac{-999}{100^2}\)

\(D=-\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\)

\(D=-\dfrac{\left(1.2.3.......99\right)\left(3.4.5...101\right)}{\left(2.3.4....100\right)\left(2.3.4......100\right)}\)

\(D=-\dfrac{1.101}{100.2}\)

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