Question

Rút gọn

\(\frac{1}{x.\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+...+\frac{1}{\left(x+2017\right)\left(x+2018\right)}\)

Answer 1

\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+....+\frac{1}{\left(x+2017\right)\left(x+2018\right)}\)

\(=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+2}-\frac{1}{x+3}+.....+\frac{1}{x+2017}-\frac{1}{x+2018}\)

\(=\frac{1}{x}-\frac{1}{x+2018}\)

Answer 2

\(\frac{1}{x.\left(x+1\right)}\)+ \(\frac{1}{\left(x+1\right)\left(x+2\right)}\)+ . . . + \(\frac{1}{\left(x+2017\right)\left(x+2018\right)}\)

= \(\frac{1}{x}\)+ \(\frac{1}{x+1}\)+ \(\frac{1}{x+2}\)- \(\frac{1}{x+3}\)+ . . . + \(\frac{1}{x+2017}\)- \(\frac{1}{x+2018}\)

= \(\frac{1}{x}\)- \(\frac{1}{x+2018}\)