You definitely don't need to know any of that background to be able to arrive at the answer. To fully understand everything maybe, but all it takes is:
The question doesn’t ask for an answer, it asks for a proof. You can’t just write a bunch of algebra and call it a day. You have to justify all of your arguments.
a = x^1 + x^4 + x^7 + ... = x(1 + x^3 + x^6 + ...) = x/(1-x^3)
a + a^3 + a^5 + ... = a(1 + a^2 + a^4 + ...) = a/(1-a^2)
Substitute + simplify. I don't think this is beyond a (fairly smart) elementary school student.