Question 56
Prove that $1$ is the only positive value of $r$ for which there is a two-way infinite sequence $a(n)$ such that
$$
a(-n) = 1 - \sum_{k=0}^{n} r^k a(n-k), \ n = 0, \pm 1, \pm 2, \dots
$$
Determine $a(n)$, $n = 0, \pm1, \pm2, \dots$, when $r = 1$.
$$
a(-n) = 1 - \sum_{k=0}^{n} r^k a(n-k), \ n = 0, \pm 1, \pm 2, \dots
$$
Determine $a(n)$, $n = 0, \pm1, \pm2, \dots$, when $r = 1$.