Question

 (1.2)² / ( 2.3)² . (3.4)²/ (4.5)² .... (999.1000)² / (1000.1001)²

Answer 1

\(\frac{\left(1.2\right)^2}{\left(2.3\right)^2}.\frac{\left(3.4\right)^2}{\left(4.5\right)^2}...\frac{\left(999.1000\right)^2}{\left(1000.1001\right)^2}\)

\(=\frac{1^2.2^2}{2^2.3^2}.\frac{3^2.4^2}{4^2.5^2}...\frac{999^2.1000^2}{1000^2.1001^2}\)

\(=\frac{1^2.2^2.3^2.4^2...999^2.1000^2}{2^2.3^2.4^2.5^2...1000^2.1001^2}\)

\(=\frac{1^2}{1001^2}\)

\(=\frac{1}{1001^2}\)

Answer 2

\(\frac{\left(1.2\right)^2}{\left(2.3\right)^2}.....\frac{\left(999.1000\right)^2}{\left(1000.1001\right)^2}\)

\(=\frac{1^2.2^2}{2^2.3^2}.....\frac{999^2.1000^2}{1000^2.1001^2}\)

\(=\frac{1^2}{3^2}.\frac{3^2}{5^2}.....\frac{999^2}{1001^2}\)

\(=\frac{1^2}{1001^2}=\frac{1}{1002001}\)

Answer 3

\(\frac{1^2.2^2}{2^2.3^2}.\frac{3^2.4^2}{4^2.5^2}....\frac{999^2.1000^2}{1000^2.1001^2}=\frac{1^2}{3^2}.\frac{3^2}{5^2}...\frac{999^2}{.1001^2}=\frac{1^2}{1001^2}\)