Question

1/2+1/6+1/12....+1/x(x+1)=2015/2016

Answer 1

1/1x2+1/2x3+...+1/x(x+1)=2015/2016

1/1-1/2+1/2-1/3+...+1/x-1/x+1=2015/2016

2/1-1/x+1=2015/2016

2016/2016-1/x+1=2015/2016

1/x+1=2016/2016-2015/2016

1/x+1=1/2016

x+1=2016

x=2016-1

x=2015

Answer 2

         \(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{x\left(x+1\right)}=\frac{2015}{2016}.\)

 <=>\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x\left(x+1\right)}=\frac{2015}{2016}\)

 <=> \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{2015}{2016}\)

 <=> \(-\frac{1}{x+1}=\frac{-1}{2016}\) <=>  x+1 = 2016  <=> x = 2015