Question

\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{x\left(x+1\right)}=\frac{2009}{2010}\)

Answer 1

\(\frac{1}{1.2}+\frac{1}{2.3}+................+\frac{1}{x.\left(x+1\right)}\)\(=\frac{2009}{2010}\)

\(\Rightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.............+\frac{1}{x}-\frac{1}{x+1}=\frac{2009}{2010}\)

\(\Rightarrow1-\frac{1}{x+1}=\frac{2009}{2010}\)

\(\Rightarrow\frac{1}{x+1}=1-\frac{2009}{2010}\)

\(\Rightarrow\frac{1}{x+1}=\frac{1}{2010}\)

\(\Rightarrow x+1=2010\)

\(\Rightarrow x=2009\)

Answer 2

x=2009

k mik nha

Answer 3

Ta thấy: \(\frac{1}{1.2}=\frac{1}{1}-\frac{1}{2}\)

Tương tự: \(\frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\)

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

Vậy: \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{x\left(x+1\right)}=\frac{1}{1}-\frac{1}{x+1}=\frac{2009}{2010}\)

<=> \(\frac{x+1-1}{x+1}=\frac{2009}{2010}\Leftrightarrow\frac{x}{x+1}=\frac{2009}{2010}\)

=> x = 2009