Functional Equation's final step

Question

I am pretty close on solving a functional equation but i am stuck on $f(f(x +2017)-f(x+1))=constant$ ,for all real x. Should I be able to get something out of it or should I change direction? I am stuck for hours. Post note: the constant is non-zero and $f(0)=0$

Answer 1

There are linear solutions to this where $f(x) = mx +b $. Given that we know $f(0) = 0$ then $f(x) = mx$.

Taking $f(f(x+2017) - f(x+1)) = k$ we can substitute $$\begin{align}f(m(x+2017) - m(x +1)) &= k \\ f(mx+2017m - mx -m) &= \\ f(2016m) &= \\ 2016m^2 &= \\ m &= \pm \sqrt{\frac{k}{2016}} \end{align}$$