Question

I=\(\frac{2^2}{1.3}\).\(\frac{3^2}{2.4}\).\(\frac{4^2}{3.5}\).\(\frac{5^2}{4.6}\).................\(\frac{999^2}{998.1000}\)=?​​​​​​​​​​

Answer 1

\(I=\frac{2^2}{1.3}.\frac{3^2}{2.4}.\frac{4^2}{3.5}...\frac{999^2}{998.1000}\)

\(I=\frac{2^2.3^2.4^2...999^2}{2.3^2.4^2...998^2.1000}\)

\(I=\frac{2}{1000}=\frac{1}{500}\)

Answer 2

\(=\frac{2\cdot2}{1\cdot3}\cdot\frac{3\cdot3}{2\cdot4}\cdot\frac{4\cdot4}{3\cdot5}\cdot...\cdot\frac{999\cdot999}{998\cdot1000}\)

\(=\frac{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot999\cdot999}{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot998\cdot1000}\)

\(=\frac{2\cdot3\cdot4\cdot...\cdot999}{1\cdot2\cdot3\cdot...\cdot998}\cdot\frac{2\cdot3\cdot4\cdot...\cdot999}{3\cdot4\cdot5\cdot...\cdot1000}\)

\(=\frac{999}{1}\cdot\frac{2}{1000}\)

\(=\frac{999}{500}\)