Question

Tính

1/3+1/6+1/10+...+2/n(n+1)=2015/2016

Answer 1

\(\Leftrightarrow\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{20}+...+\dfrac{2}{n\left(n+1\right)}=\dfrac{2015}{2016}\)

\(\Leftrightarrow2\left(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{n\left(n+1\right)}\right)=\dfrac{2015}{2016}\)

\(\Leftrightarrow2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{n}-\dfrac{1}{n+1}\right)=\dfrac{2015}{2016}\)

\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{n+1}=\dfrac{2015}{4032}\)

\(\Leftrightarrow\dfrac{1}{n+2}=\dfrac{1}{4032}\)

hay n=4030