I am trying to prove the following proposition by induction. I got stuck . Please help.
$$uf(n)+f(n-1)=g(n)\equiv 0\pmod x$$ where $u\in\mathbb Z_{\ge 0}$, and $n\in\{1,2,\cdots,p-2\}$.
If $f(0)\equiv 0\pmod x$ and $g(0)\equiv 0\pmod x$ then $f(n)\equiv 0\pmod x$.
Proof
What I plan to do:
Show that $f(1)$ is true.
Assume for some $k\in\{0,1,2,p-3\}$ that $f(k)$ is true.
Show that implies that $f(k+1)$ is necessarily true.
I got stuck from the get-go:
$uf(1)+f(0)=g(0)\equiv 0\pmod x$ $\implies$ $$uf(1)\equiv 0\pmod x$$ But how do I prove that $$f(1)\equiv 0\pmod x?$$
Any help will be greatly appreciated.Thanks