Question

[Phạm Thị Minh Ngọc]

1/3+1/6+1/10+..+2/x(x+1) = 2/2013

tìm x

Answer 1

[shitbo]

\(=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{x\left(x+1\right)}=2\left(\frac{1}{2}-\frac{1}{3}+...-\frac{1}{x+1}\right)\)

\(=2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2x-2}{2x+2}=\frac{2}{2013}\left(\text{vô nghiệm}\right);\frac{1}{3}>\frac{2}{2013}\text{ do đó vô nghiệm}\left(\text{ngắn hơn :))}\right)\)

Answer 2

[Khánh Ngọc]

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

\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2}{2013}\)

\(\Rightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2}{2013}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2}{2013}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2}{2013}\)

\(\Rightarrow\frac{2x-2}{2x+2}=\frac{2}{2013}\)

\(\Rightarrow\frac{x-1}{x+1}=\frac{2}{2013}\left(vl\right)\)

=> Bt trên có x vô nghiệm