Question

Answer 1

c: \(=\dfrac{-1}{\left(x-1\right)\left(x-2\right)}+\dfrac{-1}{\left(x-2\right)\left(x-3\right)}+...+\dfrac{-1}{\left(x-99\right)\left(x-100\right)}\)

\(=-\left(\dfrac{1}{\left(x-1\right)\left(x-2\right)}+\dfrac{1}{\left(x-2\right)\left(x-3\right)}+...+\dfrac{1}{\left(x-99\right)\left(x-100\right)}\right)\)

\(=-\left(\dfrac{-1}{x-1}+\dfrac{1}{x-2}-\dfrac{1}{x-2}+\dfrac{1}{x-3}-...-\dfrac{1}{x-99}+\dfrac{1}{x-100}\right)\)

\(=\dfrac{1}{x-1}-\dfrac{1}{x-100}\)

\(=\dfrac{x-100-x+1}{\left(x-1\right)\left(x-100\right)}=\dfrac{-99}{\left(x-1\right)\left(x-100\right)}\)