Ta có:
\(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+.....+\dfrac{2}{x\left(x+1\right)}=\dfrac{998}{1000}\)
\(\Rightarrow\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{20}+.....+\dfrac{2}{x.\left(x+1\right)}=\dfrac{998}{1000}\)
\(\Rightarrow\dfrac{2}{2.3}+\dfrac{2}{3.4}+\dfrac{2}{4.5}+.....+\dfrac{2}{x\left(x+1\right)}=\dfrac{998}{1000}\)
\(2.\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+......+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{998}{1000}\)
\(2.\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+.....+\dfrac{1}{x}-\dfrac{1}{x+1}\right)=\dfrac{998}{1000}\)
\(2.\left(\dfrac{1}{2}-\dfrac{1}{x+1}\right)=\dfrac{998}{1000}\)
\(\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{998}{1000}:2\)
\(\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{499}{1000}\)
\(\dfrac{1}{x+1}=\dfrac{1}{2}-\dfrac{499}{1000}\)
\(\dfrac{1}{x+1}=\dfrac{1}{1000}\)
\(\Rightarrow x+1=1000\)
x = 1000 - 1
x = 999
Vậy x = 999 là giá trị cần tìm
