Question

Tính \(\frac{1}{x\left(x+1\right)\left(x+2\right)}\)+ \(\frac{1}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}\)+..............+\(\frac{1}{\left(x+8\right)\left(x+9\right)\left(x+10\right)}\)

Answer 1

\(A=\frac{1}{x.\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}+...+\frac{1}{\left(x+8\right)\left(x+9\right)\left(x+10\right)}\)

\(\Leftrightarrow2A=\frac{2}{x.\left(x+1\right)\left(x+2\right)}+\frac{2}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}+...+\frac{2}{\left(x+8\right)\left(x+9\right)\left(x+10\right)}\)

\(=\frac{1}{x\left(x+1\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}+...+\frac{1}{\left(x+8\right)\left(x+9\right)}-\frac{1}{\left(x+9\right)\left(x+10\right)}\)

\(=\frac{1}{x\left(x+1\right)}-\frac{1}{\left(x+9\right)\left(x+10\right)}\)

\(\Rightarrow A=\frac{1}{2}.\left(\frac{1}{x\left(x+1\right)}-\frac{1}{\left(x+9\right)\left(x+10\right)}\right)\)