Answer:
The given geometric sequence has the first term \(a_1 = \frac{2}{3}\) and the recurrence relation \(a_n = 9 \cdot a^{n-1}\).
To find the explicit rule, we can express the recurrence relation in terms of the first term \(a_1\). The general form of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(r\) is the common ratio.
In this case, the common ratio (\(r\)) is 9:
\[ a_n = \left(\frac{2}{3}\right) \cdot 9^{n-1} \]
Now, let's simplify the expression:
\[ a_n = \frac{2 \cdot 9^{n-1}}{3} \]
The correct option is:
a. \( a_n = 9 \cdot \left(\frac{2}{3}\right)^n \)