Question

(1/2^2-1)(1/3^2-1)(1/4^2-1)...(1/2023^2-1)

Answer 1

\(=\dfrac{1-2^2}{2^2}\cdot\dfrac{1-3^2}{3^2}\cdot...\cdot\dfrac{1-2023^2}{2023^2}\)

\(=\dfrac{2^2-1}{2^2}\cdot\dfrac{3^2-1}{3^2}\cdot...\cdot\dfrac{2023^2-1}{2023^2}\)

\(=\dfrac{1}{2}\cdot\dfrac{3}{2}\cdot\dfrac{2}{3}\cdot\dfrac{4}{3}\cdot...\cdot\dfrac{2022}{2023}\cdot\dfrac{2024}{2023}\)

\(=\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot...\cdot\dfrac{2022}{2023}\cdot\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot...\cdot\dfrac{2024}{2023}\)

\(=\dfrac{1}{2023}\cdot\dfrac{2024}{2}=\dfrac{1012}{2023}\)

Answer 2

=1012/2023