Question

tính: A = \(\left(\dfrac{1}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right).....\left(\dfrac{1}{100^2}-1\right)\)

giup mk nha

Answer 1

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

\(=\left(\dfrac{1}{4}-\dfrac{4}{4}\right).\left(\dfrac{1}{9}-\dfrac{9}{9}\right)..............\left(\dfrac{1}{100^2}-\dfrac{10000}{10000}\right)\)

\(=\dfrac{-3}{4}.\dfrac{-8}{9}...............\dfrac{-9999}{10000}\)

\(\dfrac{\left(-1\right).3}{2^2}.\dfrac{\left(-2\right).4}{3^2}................\dfrac{\left(-99\right).101}{100^2}\)

\(=\dfrac{\left(-1\right).\left(-2\right)............\left(-99\right)}{2.3...100}.\dfrac{3.4.5.....101}{2.3.....100}\)

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

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