Question

F(x) = g(x+1)

Answer 1

$g(x)=h(2x)-x\implies h(2x)=g(x)+x$

$f(x)=g(x+1)\implies g(x)=f(x-1)$

Taken together, we get

$h(2x)=f(x-1)+x$

Since $2018=2\cdot1009$, the above gives

$a^3=f(1008)+1009$

We have

$f(y)=\displaystyle\sum_{i=0}^{1008}(2018-2i)y=1019090y$

so that

$a^3=1019090\cdot1008+1009$

$a^3=1027243729=1009^3$

$\implies a=1009$