Recursive Definitions - Should I use logic notation (i.e. arrows or "for n =...")

Question

[b.] $a_n = 1 + (-1)^n $

\begin{align*} a_1 = 1 + (-1)^1 = 0\\ a_2 = 1 + (-1)^2 = 2\\ a_3 = 1 + (-1)^3 = 0\\ a_4 = 1 + (-1)^4 = 2\\ \vdots \\ \\ \text{Recurisve Definition: }\\ a_1 = 0 \\ a_n = 2 \text{ for } a_{n-1} = 0 \\ a_n = 0 \text{ for } a_{n-1} = 2 \\ \end{align*}

Should it be like that ^

Or like this: \begin{align*} \text{Recurisve Definition: }\\ a_1 = 0 \\ a_{n-1} = 0 \rightarrow a_n = 2 \\ a_{n-1} = 2 \rightarrow a_n = 0 \\ \end{align*}

Answer 1

I don't think any of those is really standard. One thing you might write is:

$$ \begin{align*} a_1 & = 0 \\ a_n & = \begin{cases} 2,&\text{when $a_{n-1} = 0$} \\ 0,&\text{when $a_{n-1} = 2$} \end{cases} \end{align*}$$

But more likely:

$$ \begin{align*} a_1 & = 0 \\ a_n & = a_{n-1} + 2\cdot(-1)^n & (n>1) \end{align*}$$

But it's a little hard to say because the problem is so contrived. Who needs a recursive definition of $a_n = 1 + (-1)^n$ ?