Question

\(\left(1+\dfrac{2}{2.3}\right).\left(1+\dfrac{2}{3.4}\right).....\left(1+\dfrac{2}{n\left(n+1\right)}\right)\)

Answer 1

\(\Rightarrow\left(1+1+...+1\right)+2\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+...\dfrac{1}{n\left(n+1\right)}\right)\)[có (n-1) số 1]

\(\Rightarrow\left(n-1\right)+2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{n}-\dfrac{1}{n+1}\right)\)

\(\Rightarrow\left(n-1\right)+2\left(\dfrac{1}{2}-\dfrac{1}{n+1}\right)\)

\(\Rightarrow\left(n-1\right)+\left(1-\dfrac{2}{n+1}\right)\)

\(\Rightarrow n-\dfrac{2}{n+1}\)

\(\Rightarrow\dfrac{n\left(n+1\right)}{n+1}-\dfrac{2}{n+1}\)

\(\Rightarrow\dfrac{n^2+n-2}{n+1}\)