Question

[park jimin]

1/3+1/6+1/10+...+1/x(x+1)=1010/1012

Answer 1

[Umi]

đề sai 1/x(x + 1) phải là 2/x(x + 1)

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{1010}{1012}\)

\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{1010}{1012}\)

\(\Rightarrow2\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{1010}{1012}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{1010}{1012}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{1010}{1012}\)

\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{505}{1012}\)

\(\Rightarrow\frac{1}{x+1}=\frac{1}{1012}\)

\(\Rightarrow x+1=1012\)

\(\Rightarrow x=1011\)