How do I use induction to show this? From what I understand we want to prove $\forall m \in \mathbb{N}$, $$ m = \sum_{i=0}^{n}{a_i.3^i} = a_0.3^0 + a_1.3^1 + a_2.3^2 + \dots + a_{n-1}.3^{n-1} + a_n.3^n $$ where $n \geq 0$ is unique and so is $a_i$, for $a_i = 2, 1$ or $0$.
The base case is simple for $m=1$: Let $a_0 = 1$. Then $1 = 1.3^0$ which is true. How do I proceed after assuming the inductive hypothesis? Adding $1$ to show the expression for $m$ to show the result for $m+1$ seems a bit trivial..
Hint: every non-negative integer $m$ can be expressed uniquely in the form: $$m=r+3m_1$$ Where $r=0,1$ or $2$ and $m_1$ is a non-negative integer.
You need to use "course of values" induction. I.e. to show it holds for $m$ if it holds for ALL $0 \le m_1<m$.