Question

\(\left(1+\frac{1}{1\cdot3}\right)\cdot\left(1+\frac{1}{2\cdot4}\right)\cdot\cdot\cdot\cdot\left(1+\frac{1}{2004\cdot2006}\right)\)

Answer 1

\(\left(1+\frac{1}{1\cdot3}\right)\left(1+\frac{1}{2\cdot4}\right)\left(1+\frac{1}{3\cdot5}\right)...\left(1+\frac{1}{2004\cdot2006}\right)\)

\(=\frac{4}{1\cdot3}+\frac{9}{2\cdot4}+\frac{16}{3\cdot5}+...+\frac{420025}{2004\cdot2006}\)

\(=\frac{\left(2\cdot2\right)\left(3\cdot3\right)\left(4\cdot4\right)...\left(2005\cdot2005\right)}{\left(1\cdot3\right)\left(2\cdot4\right)\left(3\cdot5\right)...\left(2004\cdot2006\right)}\)

\(=\frac{\left(2\cdot3\cdot4\cdot...\cdot2005\right)\left(2\cdot3\cdot4\cdot...\cdot2005\right)}{\left(1\cdot2\cdot3\cdot...\cdot2004\right)\left(3\cdot4\cdot5\cdot...\cdot2006\right)}\)

\(=\frac{2005\cdot2}{1\cdot2006}\)

\(=\frac{4010}{2006}\)

Answer 2

\(\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)...\left(1+\frac{1}{2004.2006}\right)\)

\(=\frac{1.3+1}{1.3}.\frac{2.4+1}{2.4}....\frac{2004.2006+1}{2004.2006}\)

\(=\frac{2^2}{1.3}.\frac{3^2}{2.4}....\frac{2005^2}{2004.2006}\)

\(=\frac{2.3....2005}{1.2....2004}.\frac{2.3...2005}{3.4....2006}\)

\(=2005.\frac{2}{2006}=\frac{2005}{1003}\)

Answer 3

\(=\left(1+\frac{1}{1.3}\right).\left(1+\frac{1}{2.4}\right)...\left(1+\frac{1}{2004.2006}\right)\)

\(=\left(\frac{2.2}{1.3}\right).\left(\frac{3.3}{2.4}\right)....\left(\frac{2005.2005}{2004.2006}\right)\)

\(=\frac{\left(2.3.4....2005\right).\left(2.3.4....2005\right)}{\left(1.2.3...2004\right).\left(3.4.5....2006\right)}\)

\(=\frac{2005}{2006}\)