Question

Tìm x, biết:

\(\frac{1}{x\left(x+1\right)}\)+  \(\frac{1}{\left(x+1\right)\left(x+2\right)}\)+ ... + \(\frac{1}{\left(x+99\right)\left(x+100\right)}\)=  \(\frac{100}{101}\)

 

Answer 1

\(\frac{\left(x+1\right)-x}{x\left(x+1\right)}+\frac{\left(x+2\right)-\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}+...+\frac{\left(x+100\right)-\left(x+99\right)}{\left(x+99\right)\left(x+100\right)}=\frac{100}{101}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+...+\frac{1}{x+99}-\frac{1}{x+100}=\frac{100}{101}\)
\(\Leftrightarrow\frac{1}{x}-\frac{1}{x+100}=\frac{100}{101}\)
Tự giải nha

Answer 2

1/x -1/x+100 = 100/101

Answer 3

\(\frac{1}{x}-\frac{1}{x+100}=\frac{100}{101}\)
\(\Leftrightarrow\frac{x+100-x}{x\left(x+100\right)}=\frac{100}{101}\)
\(\Leftrightarrow\frac{100}{x\left(x+100\right)}=\frac{100}{101}\)
\(\Leftrightarrow x\left(x+100\right)=101\)
\(\orbr{\begin{cases}x=1\\x=-101\end{cases}}\)