On: Discrete Mathematics: An Open Introduction [pdf]

[chongli]

And the idea of a formal power series. And integer compositions. And combinatorial enumeration (counting sets in different ways for a proof). And a bit of set theory (cardinality of sets).

There is a whole lot of background stuff here that elementary school students do not have. Way more than what you’ve stated.

[rak1507 → chongli]

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:

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.

[chongli → rak1507]

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.

[rak1507 → chongli]

There aren't really any complicated arguments being made, so I don't think a proof would be that involved.