`1/(x(x + 1)) + 1/((x + 1)(x + 2)) + 1/((x + 2)(x + 3)) - 1/x = 1/2017`
`⇒ ((x - 1) - x)/(x(x + 1)) + ((x + 2) - (x + 1))/((x + 1)(x + 2)) + ((x + 3) - (x + 2))/((x + 2)(x + 3)) - 1/x = 1/2017`
`⇒ 1/x - 1/(x + 1) + 1/(x + 1) - 1/(x + 2) + 1/(x + 2) - 1/(x + 3) - 1/x = 1/2017`
`⇒ 1/x - 1/(x + 3) - 1/x = 1/2017`
`⇒ -1/(x + 3) =1/2017`
`⇒ x + 3 = -2017`
`⇒ x = -2020`