Question

Tính A= (1/2^2-1)(1/3^2-1)...(1/100^2-1)

Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right)-...+\left(\dfrac{1}{99^2}-1\right).\left(\dfrac{1}{100^2}-1\right)\)

\(A=\left(\dfrac{1}{2}+1\right).\left(\dfrac{1}{2}-1\right).\left(\dfrac{1}{3}-1\right).\left(\dfrac{1}{3}+1\right)+...\left(\dfrac{1}{99}+1\right).\left(\dfrac{1}{99}-1\right)\)

\(A=\dfrac{-3}{2}.\dfrac{-1}{2}.\dfrac{-2}{3}.\dfrac{4}{3}+...+\dfrac{100}{99}.\dfrac{98}{100}.\dfrac{-99}{100}.\dfrac{101}{100}\)

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

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