Solution
= Solution
Let $p_n=\mathbb P(X_n=A)$. Whenever the taxi is away from the airport it returns with probability $3/4$, whereas it must leave whenever it is at the airport. Hence
$$
p_{n+1}=\frac34(1-p_n),
\qquad p_0=1.
$$
Subtracting the fixed point $3/7$ gives
$$
p_{n+1}-\frac37=-\frac34
\left(p_n-\frac37\right).
$$
Therefore
$$
\boxed{
p_n=\frac37+\frac47\left(-\frac34\right)^n
}
$$
for every $n\geq0$, and in particular for the requested $n\geq1$.