Question

\(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{\left(5x+1\right)\left(5x+6\right)}=\dfrac{2015}{2016}\)

Answer 1

\(\dfrac{5}{1.6}+\dfrac{5}{6.11}+...+\dfrac{5}{\left(5x+1\right)\left(5x+6\right)}=\dfrac{2015}{2016}\)

\(\Rightarrow\dfrac{1}{1}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{5x+1}-\dfrac{1}{5x+6}=\dfrac{2015}{2016}\)

\(\Rightarrow\dfrac{1}{1}-\dfrac{1}{5x+6}=\dfrac{2015}{2016}\)

\(\Rightarrow\dfrac{5x+5}{5x+6}=\dfrac{2015}{2016}\)

\(\Rightarrow\left(5x+5\right).2016=\left(5x+6\right).2015\)

\(\Rightarrow10080x+10080=10075+12090\)

\(\Rightarrow5x=2010\)

\(\Rightarrow x=402\)

Vậy x = 402