Question

[Nguyễn Thị Thảo Nhi]

thực hiện phép tính 1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+....+1/(x+2013)(x+2014)

Answer 1

[Trâm Nguyễn Thị Ngọc]

Đặt biểu thức là A

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

\(\Leftrightarrow\dfrac{x+1}{x\left(x+1\right)}-\dfrac{x}{x\left(x+1\right)}+\dfrac{x+2}{\left(x+1\right)\left(x+2\right)}-\dfrac{x+1}{\left(x+1\right)\left(x+2\right)}+...+\dfrac{x+2014}{\left(x+2013\right)\left(x+2014\right)}-\dfrac{x+2013}{\left(x+2013\right)\left(x+2014\right)}\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}-\dfrac{1}{x+2}-...-\dfrac{1}{x+2013}+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}.\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+2014}\)

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

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