\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{x\left(x+1\right)}=\frac{996}{997}\)
\(\Rightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{996}{997}\)
\(\Rightarrow1-\frac{1}{x+1}=\frac{996}{997}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{997}\)
\(\Rightarrow x+1=997\)
\(\Rightarrow x=996\)
\(\Leftrightarrow\)1-1/2+1/2-1/3+1/3-1/4+..+1/x-1/(x+1)=996/997
\(\Leftrightarrow\)1-1/(x+1)=996/997
\(\Leftrightarrow\)\(\frac{x}{x+1}\)\(=\frac{996}{997}\)
\(\Leftrightarrow x=996\)
\(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{996}{997}\)
Xét dạng tổng quát : \(\frac{1}{a}-\frac{1}{a+1}=\frac{a+1}{ax\left(a+1\right)}-\frac{a}{ax\left(a+1\right)}=\frac{a+1-a}{ax\left(a+1\right)}=\frac{1}{ax\left(a+1\right)}\)
Do đó \(\frac{1}{ax\left(a+1\right)}=\frac{1}{a}-\frac{1}{a+1}\)
Áp dụng \(\frac{1}{1x2}=\frac{1}{1x\left(1+1\right)}=\frac{1}{1}-\frac{1}{1+1}=\frac{1}{1}-\frac{1}{2}\)
\(\frac{1}{2x3}=\frac{1}{2x\left(2+1\right)}=\frac{1}{2}-\frac{1}{2+1}=\frac{1}{2}-\frac{1}{3}\)
.......
\(\frac{1}{Xx\left(X+1\right)}=\frac{1}{X}-\frac{1}{X+1}\)
Do đó \(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{X}-\frac{1}{X+1}\)
\(=\frac{1}{1}-\frac{1}{X+1}=1-\frac{1}{X+1}=\frac{X+1}{X+1}-\frac{1}{X+1}=\frac{X}{X+1}\)
Vì \(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{996}{997}\)
nên \(\frac{X}{X+1}=\frac{996}{997}\)
\(\frac{X}{X+1}=\frac{996}{996+1}\)
Vậy X=996