$A=\{1\}$, $A_n=\{3k, 3k+1\}$, where $k$ is in $A_{n-1}$. Let $A$ is the union of $A_n$ where $n=1,2,3,...$. Can $2017$ be written as a sum of two elements of $A$? Is this representation unique?
I have tried this. And with a little calculation I showed that $A$ contains elements of the form $3^n, 3^n+1, 3^n+3^{n-1}+1, 3^n+3^{n-1}+3^2+1$ etc. And with this I was unable to express $2017$ as a sum of two elements of $A$. So I came to the conclusion that $2017$ can't be expressed as a sum. But I don't think this is true. So how to solve it?
$A$ is the smallest set containing $1$, and closed under the operations $n\mapsto3n$ and $n\mapsto3n+1$. It follows that $A$ contains all natural numbers whose base $3$ representations contains just the digits $0$ and $1$ (and not $2$). Taking the sum of two of these numbers will give rise to no carries in base $3$. I would suggest looking at the base $3$ representation of $2017$.