= Solution
Assume $n\geq3$ and use periodic indices for the <Jacobi method>. At iteration $r$, it is
$$
\boxed{x_j^{(r+1)}=\tfrac12\bigl(x_{j-1}^{(r)}+x_{j+1}^{(r)}-b_j\bigr),\qquad x_0=x_n,\quad x_{n+1}=x_1.}
$$
All values on its right side belong to the old iterate. For lexicographic <Gauss-Seidel iteration>, compute in the order $1,\ldots,n$ and use a newly available coordinate immediately:
$$
\begin{aligned}
x_1^{(r+1)}&=\tfrac12(x_2^{(r)}+x_n^{(r)}-b_1),\\
x_j^{(r+1)}&=\tfrac12(x_{j-1}^{(r+1)}+x_{j+1}^{(r)}-b_j),\quad 2\leq j\leq n-1,\\
x_n^{(r+1)}&=\tfrac12(x_1^{(r+1)}+x_{n-1}^{(r+1)}-b_n).
\end{aligned}
$$
The wrap-around term in the last row uses the new $x_1$, while the wrap-around term in the first row uses the old $x_n$. This is important for the actual <iteration matrix> and its convergence behavior.
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