Question

Tính nhanh:

A= 1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+...+1/(x+2009)(x+2010)

Answer 1

A= \(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+....\frac{1}{\left(x+2009\right)\left(x+2010\right)}\)

\(A=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x-1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}...\frac{1}{x+2009}-\frac{1}{x+2010}\)

\(A=\frac{1}{x}-\frac{1}{x+2010}\)

\(A=\frac{1}{x}+\frac{-1}{x+2010}\)

\(A=\frac{1\left(x+2010\right)}{x\left(x+2010\right)}+\frac{-1\cdot x}{x\left(x+2010\right)}\)

\(A=\frac{x+2010-x}{x\left(x+2010\right)}=\frac{2010}{x\left(x+2010\right)}\)