Question

\[ C = \left( 1 + \frac{4}{1.5} \right) \left( 1 + \frac{4}{2.6} \right) \left( 1 + \frac{4}{3.7} \right) \ldots \left( 1 + \frac{4}{96.100} \right) \times \frac{99}{97} \]

Answer 1

\(1+\dfrac{4}{a\left(a+4\right)}=\dfrac{a\left(a+4\right)+4}{a\left(a+4\right)}=\dfrac{\left(a+2\right)^2}{a\left(a+4\right)}\)

\(C=\left(1+\dfrac{4}{1\cdot5}\right)\left(1+\dfrac{4}{2\cdot6}\right)\cdot...\cdot\left(1+\dfrac{4}{96\cdot100}\right)\cdot\dfrac{99}{97}\)

\(=\dfrac{\left(1+2\right)^2}{1\left(1+4\right)}\cdot\dfrac{\left(2+2\right)^2}{2\left(2+4\right)}\cdot...\cdot\dfrac{\left(96+2\right)^2}{96\left(96+4\right)}\cdot\dfrac{99}{97}\)

\(=\dfrac{3\cdot4\cdot...\cdot98}{1\cdot2\cdot...\cdot96}\cdot\dfrac{3\cdot4\cdot...\cdot98}{5\cdot6\cdot...\cdot100}\cdot\dfrac{99}{97}\)

\(=\dfrac{97\cdot98}{2}\cdot\dfrac{3\cdot4}{99\cdot100}\cdot\dfrac{99}{97}=\dfrac{97}{99}\cdot\dfrac{99}{97}\cdot\dfrac{98\cdot3\cdot4}{2\cdot100}\)

\(=\dfrac{98\cdot3\cdot2}{100}=\dfrac{4\cdot49\cdot3}{100}=\dfrac{49\cdot3}{25}=\dfrac{147}{25}\)