Question

Tính

\(A=\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+.....+\dfrac{1}{\left(x+2013\right)\left(x+2014\right)}\)

Answer 1

\(A=\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+...+\dfrac{1}{\left(x+2013\right)\left(x+2014\right)}\)

\(\Rightarrow A=\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+...+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}\)

\(\Rightarrow A=\dfrac{1}{x}-\dfrac{1}{x+2014}\)

\(\Rightarrow A=\dfrac{2014}{x\left(x+2014\right)}\)

Answer 2

\(A=\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+....+\dfrac{1}{\left(x+2013\right)\left(x+2014\right)}\)

\(=\dfrac{1}{x}+\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+\dfrac{1}{x+2}-\dfrac{1}{x+3}+...+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}\)

\(=\dfrac{1}{x}-\dfrac{1}{x+2014}-\dfrac{x+2014}{x\left(x+2014\right)}-\dfrac{x}{x\left(x+2014\right)}\)

\(=\dfrac{x+2014-x}{x\left(x+2014\right)}\)

\(=\dfrac{2014}{x\left(x+2014\right)}\)