Question

1/6+1/12+1/20+1/30+1/42+........+1/x(x+1)=1005/2012

Answer 1

S = 1/2·3 + 1/3·4 + 1/4·5 + 1/5·6 + 1/6·7 + ............+ 1/x(x+1) = (1/2 - 1/3 +1/3 -1/4 +1/4 - 1/5 +...............+ 1/x - 1/ (x+1) =
1/2 - 1/(x+1) = (x-1)/2(x+1)
(x-1)/2(x+1) = 1005/2012
1006(x-1) = 1005(x+1) 
x= 1005+1006 = 2011