Let $P_k$ denote the probability that code $A$ is used in the $k$\text{th}$week. Then the probability that$A$is not used in the$k$\text{th}$ week is $1 - P_k$. Therefore, we have
$$P_{k+1} = \frac{1}{3}(1 - P_k).$$
Or
$$P_{k+1} - \frac{1}{4} = -\frac{1}{3}\left(P_k - \frac{1}{4}\right).$$
As $P_1 = 1$, $\{P_k - \frac{1}{4}\}$ is then a geometric sequence with $\frac{3}{4}$ as the first term and $-\frac{1}{3}$ as the common ratio. So we have
$$P_k - \frac{1}{4} = \frac{3}{4}\left(-\frac{1}{3}\right)^{k-1}.$$
Or
$$P_k = \frac{3}{4}\left(-\frac{1}{3}\right)^{k-1} + \frac{1}{4}.$$
Therefore, $P_7 = \frac{61}{243}$.