Question

B=(1+1/1 x 3) x (1+1/2 x 4) x (1+1/3 x 5) x..x (1+1/99 x 101)

Answer 1

\(B=\left(1+\dfrac{1}{1\cdot3}\right)\left(1+\dfrac{1}{2\cdot4}\right)\cdot...\cdot\left(1+\dfrac{1}{99\cdot101}\right)\)

\(=\left(1+\dfrac{1}{2^2-1}\right)\left(1+\dfrac{1}{3^2-1}\right)\cdot...\cdot\left(1+\dfrac{1}{100^2-1}\right)\)

\(=\dfrac{2^2}{\left(2-1\right)\left(2+1\right)}\cdot\dfrac{3^2}{\left(3-1\right)\left(3+1\right)}\cdot...\cdot\dfrac{100^2}{\left(100-1\right)\left(100+1\right)}\)

\(=\dfrac{2\cdot3\cdot...\cdot100}{1\cdot2\cdot...\cdot99}\cdot\dfrac{2\cdot3\cdot...\cdot100}{3\cdot4\cdot...\cdot101}\)

\(=\dfrac{100}{1}\cdot\dfrac{2}{101}=\dfrac{200}{101}\)