Question

Tìm x biết

\(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{2011}{2013}\)

Answer 1

=> \(\dfrac{2}{6}\)+\(\dfrac{2}{12}\)+\(\dfrac{2}{20}\)+...+\(\dfrac{2}{x.\left(x+1\right)}\)=\(\dfrac{2011}{2013}\)

=> 2.(\(\dfrac{1}{2.3}\)+\(\dfrac{1}{3.4}\)+\(\dfrac{1}{4.5}\)+...+\(\dfrac{1}{x.\left(x+1\right)}\)=\(\dfrac{2011}{2013}\)

=> 2.(\(\dfrac{1}{2}\)-\(\dfrac{1}{3}\)+\(\dfrac{1}{3}\)-\(\dfrac{1}{4}\)+...+\(\dfrac{1}{x}\)-\(\dfrac{1}{x+1}\)=\(\dfrac{2011}{2013}\)

=> 2.(\(\dfrac{1}{2}\)-\(\dfrac{1}{x+1}\)) =\(\dfrac{2011}{2013}\)

=>\(\dfrac{x+1-2}{2.\left(x+1\right)}\)=\(\dfrac{2011}{2013}\)

=> \(\dfrac{x-1}{x+1}\)=\(\dfrac{2011}{2013}\)

=> 2013.(x-1) = 2011.(x+1)

=> 2013x-2013= 2011x+2011

=> 2013x -2011x= 2013+2011

=> 2x= 4024

=> x= 2012

Chúc bạn học tốt!Tick cho mk nhé