= Solution
For the <autoregressive process of order one>, the formula at $t=0$ reads $r_0=r_0$, with an empty sum. If it holds at time $t$, substitution into the recursion gives
$$
r_{t+1}=\beta\left(\beta^tr_0+\sum_{s=1}^t\beta^{t-s}\xi_s\right)+\xi_{t+1}
=\beta^{t+1}r_0+\sum_{s=1}^{t+1}\beta^{t+1-s}\xi_s.
$$
Thus <mathematical induction> proves
$$
\boxed{r_t=\beta^tr_0+\sum_{s=1}^t\beta^{t-s}\xi_s.}
$$
No restriction such as $|\beta|<1$ is needed for this finite-time identity.
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